This is a question which is somewhat “advanced” in theology and apologetics. It presumes an understanding of the Church’s teaching about venial/mortal sin.
I contend that the Catholic Church has never taught that salvation was absolutely impossible for anyone who dies in a state of mortal sin, after having attained Saving Grace (through Christian Baptism).
So, I would like to discuss the question: Does the Catholic Church teach that a Baptized person who commits unconfessed mortal sin (which would not include the intent to confess in dire circumstances where Confession was not available) has absolutely NO possibility whatsoever of salvation?
Cardinal Arinze explains below that ALL that is required for an act to be considered a Mortal Sin is consent i.e. “the moment that decision is made” to commit the act you have already committed a mortal sin.
The moment someone decides to murder someone they have committed a mortal sin even if the actual physical act is not yet committed.
I didn’t read that whole thread, but the Catechism says:
***Mortal sin is a radical possibility of human freedom, as is love itself. It results in the loss of charity and the privation of sanctifying grace, that is, of the state of grace. If it is not redeemed by repentance and God’s forgiveness, it causes exclusion from Christ’s **kingdom and the eternal death of hell, for our freedom has the power to make choices for ever, with no turning back. However, although we can judge that an act is in itself a grave offense, we must entrust judgment of persons to the justice and mercy of God. *
No one can know the depths of God’s mercy. But it would be foolish to ignore this or to say that there’s a pretty good chance of salvation. That would be leading many astray.
Just my 2 cents.
If you enter heaven with an unconfessed grave sin that that means it didn’t meet the conditions required to be mortal. OR God granted the person mercy due to extenuating circumstances. The point is, we would never know if God would be merciful until it was too late.
We are bound by God’s commandments. But in theory God could make exceptions if He wanted to (He’s God after all). But we have no idea if He actually does or if He does, when He would do that. There is no revelation regarding “exceptions.”
CCC 1022 Each man receives his eternal retribution in his immortal soul at the very moment of his death, in a particular judgment that refers his life to Christ: either entrance into the blessedness of heaven-through a purification or immediately or immediate and everlasting damnation.
CCC 1033 To die in mortal sin without repenting and accepting God’s merciful love means remaining separated from him for ever by our own free choice. This state of definitive self-exclusion from communion with God and the blessed is called “hell.”
How I understand it is that one unrepentant mortal sin is enough to send a person to hell.
But the Church teaches that perfect contrition (the guilt one feels because of injuring his relationship to the Lord) is sufficient to return to a state of grace, if sacramental confession is not possible.
Imperfect contrition is the guilt we feel because we fear the punishment.
Catholic teaching distinguishes a twofold hatred of sin; one, perfect contrition, rises from the love of God Who has been grievously offended; the other, imperfect contrition, arises principally from some other motives, such as loss of heaven, fear of hell, the heinousness of sin, etc. (Council of Trent, Sess. XIV, ch. iv de Contritione). For the doctrine of imperfect contrition see ATTRITION
Grammatically, “contrition” is a noun. “Contrite” is an adjective. One repents when they are contrite. The action is the repenting, not the contrition.
the state of feeling remorseful and penitent.
synonyms: remorse, remorsefulness, repentance, penitence, sorrow, sorrowfulness, regret, ruefulness, pangs of conscience; More
(in the Roman Catholic Church) the repentance of past sins during or after confession.
“prayers of contrition”
Yes, if one dies in a state of mortal sin they cannot be saved.
However, it is impossible to tell if someone is actually in a state of mortal sin. Just because one does an objectively evil and disordered act that is grave matter does not automatically mean that the person is guilty of mortal sin. Subjectively, there might be numerous factors - such as force of habit, compulsion or peer pressure, mental disorder or imbalance - that reduce culpability.
We live in the hope that most people deep down are not culpable even when they commit intrinsically disordered actions.
Pope Benedict XVI was of the opinion and expressed it an encyclical that ‘most’ people will end up in purgatory, with only a minority of really committed mortal sinners in hell:
"…With death, our life-choice becomes definitive—our life stands before the judge. Our choice, which in the course of an entire life takes on a certain shape, can have a variety of forms. There can be people who have totally destroyed their desire for truth and readiness to love, people for whom everything has become a lie, people who have lived for hatred and have suppressed all love within themselves. This is a terrifying thought, but alarming profiles of this type can be seen in certain figures of our own history. In such people all would be beyond remedy and the destruction of good would be irrevocable: this is what we mean by the word Hell. On the other hand there can be people who are utterly pure, completely permeated by God, and thus fully open to their neighbours—people for whom communion with God even now gives direction to their entire being and whose journey towards God only brings to fulfilment what they already are.
Yet we know from experience that neither case is normal in human life. For the great majority of people—we may suppose—there remains in the depths of their being an ultimate interior openness to truth, to love, to God. In the concrete choices of life, however, it is covered over by ever new compromises with evil—much filth covers purity, but the thirst for purity remains and it still constantly re-emerges from all that is base and remains present in the soul. What happens to such individuals when they appear before the Judge?… In this text, it is in any case evident that our salvation can take different forms, that some of what is built may be burned down, that in order to be saved we personally have to pass through “fire” so as to become fully open to receiving God and able to take our place at the table of the eternal marriage-feast…"
- Pope Emeritus Benedict ZVI, Spe Salvi, 2007
The famous Thomist Fr Reginald Garrigou-Lagrange, O.P. (1877 - 1964), consulter of the Holy Office, seemed to concur that there would be more saved than damned, although he stated that this was speculative and uncertain since we could never know either way (ie God has not revealed anything on this definitively):
"…The number of the elect is known only by God. “The Lord knoweth who are His.” The liturgy says that this number is known to Him alone.  This is reaffirmed also by St. Thomas.  The end of the world will come when the number of the elect is complete, when the succession of human generations has reached its goal.
**This number in itself is very great…
When we speak of men exclusively, we do not know, first of all, if among the worlds scattered in space the earth is the only one that is habitable**…
To conclude: some insist on the mercy of God, others on the justice of God. Neither one side nor the other gives us certitude. And the reasons of appropriateness which each invokes differ very much from the reasons of appropriateness invoked in favor of a dogma which is already certain by revelation, whereas here we are treating of a truth that is not certain.
Restricting the question to Catholics, we find the doctrine, generally held especially since Suarez, that, if we consider merely adults, the number of the elect surpasses that of the reprobate. If adult Catholics do at one time or another sin mortally, nevertheless they can arise in the tribunal of penance, and there are relatively few who at the end of life do not repent, or even refuse to receive the sacraments.
But if we are treating of all Christians, of all who have been baptized, Catholic, schismatic, Protestant, it is more probable, theologians generally say, that the great number is saved. First, the number of infants who die in the state of grace before reaching the age of reason is very great. Secondly, many Protestants, being today in good faith, can be reconciled to God by an act of contrition, particularly in danger of death. Thirdly, schismatics can receive a valid absolution…
Further, among non-Christians (Jews, Mohammedans, pagans) there are souls which are elect. Jews and Mohammedans not only admit monotheism, but retain fragments of primitive revelation and of Mosaic revelation. They believe in a God who is a supernatural rewarder, and can thus, with the aid of grace, make an act of contrition. And even to pagans, who live in invincible, involuntary ignorance of the true religion, and who still attempt to observe the natural law, supernatural aids are offered, by means known to God. These, as Pius IX says,  can arrive at salvation. God never commands the impossible. To him who does what is in his power God does not refuse grace. 
We cannot arrive at certitude in this question. It is better to acknowledge our ignorance than to discourage the faithful by a doctrine which is too rigid, to expose them to danger by a doctrine which is too superficial.
Now, ‘many’ people will likely at some point have committed some kind of act that could lead to mortal sin - ie fornication, masturbation, anything sexual in particular, for which humans tend to have a specific weakness if left to their own inordinate desires. Devout Catholics will most likely have confessed and repented of this, perhaps for instance settling down into a committed marital relationship and deeply regretting their past mistakes through partaking of Penance and Acts of Contrition. Yet many more, especially, in our irreligious age, will not have. If they are likely going to be saved, as Benedict opines, we can only presume that while they are committing objectively sinful acts their culpability has been somehow reduced - perhaps due to cultural factors arguing that this sort if behaviour is normal, healthy and to be expected from young people :shrug:
Of course one is perfectly entitled to take the opposite view from Benedict that most people will end up in hell due to these objectively disordered acts. Catholic doctrine doesn’t tell us either way. I’m more partial to the view of the Pope Emeritus, since my ‘gut’ tells me that most people are sinful but on the whole ‘decent’ and searching for the ‘good’. This doesn’t excuse their behaviour but I think they will receive a ‘lesser beating’ to use the language of the parable, especially given the rise of irreligious cultural norms. I always recall the proverb:
Proverbs 14:12 - There is a way which seemeth right unto a man, but the end thereof are the ways of death
This could be interpreted two ways: since they are well-intentioned even if objectively wrong, God will have mercy on them. Or they really will end up in ‘death’ even though they are genuinely convinced they are in the right.
However, if a person does die in a state of objective and subjective ‘grave matter,’ he/she is a mortal sinner and is definitely in hell.
The answer to your very last question is “no,” but that is not exactly the same thing as the question in the title. Let me explain a little.
The absolute condicio sine qua non for being saved is to die in friendship with God, which means fundamentally that at the moment of death one must possess sanctifying grace (the habitual indwelling of God in our souls that makes us righteous, holy, and pleasing to God).
If one commits a mortal sin, the gravest consequence of that sin is that it it is incompatible with sanctifying grace. By committing that sin, we make our souls inhospitable, so to speak, to the indwelling of God.
In order for us to be restored in friendship with God, God must take the initiative (like the shepherd who looks for the lost sheep) to bring us back. It is His grace that makes us repent. Now, the ordinary channel for receiving forgiveness of mortal sins (committed after Baptism) is sacramental confession: all Catholics are obliged to use this means, unless something renders that impossible.
Having said that, confession is not the only means to obtain forgiveness for a mortal sin. There is also the act of perfect contrition, which is repentance for our sins motivated out of love for God (as opposed to something like fear of punishment).
Putting these together, we conclude the following:
(1) If someone commits a mortal sin and refuses to repent, even on his deathbed, and he persists in this condition until the moment of death, then salvation is not possible. That is, indeed, (unfortunately) how people end up condemned.
(2) On the other hand, the mere fact that someone has committed a mortal sin and has not confessed it does not necessarily mean that he will be condemned. In order to be saved, it would be sufficient for him to make an act of perfect contrition (which—for Catholics—includes the intention of confessing his sin when he reasonably has the opportunity) before he dies. It is certainly unwise to test fate by waiting for the last possible moment, but it is possible.
(Note that the approach of death itself may make it impossible for the penitent to approach confession, so in that case, of course, he is not obligated; the act of perfect contrition is sufficient. Invincible ignorance regarding the obligation to confess one’s sins would also excuse the person.)
It is very important to recall that God is infinitely merciful, and only He knows what is going on in people’s souls. It is possible that even a well catechized, baptized Catholic may be aware of the obligation to confess his sins, but be psychologically unable to do so. God knows whether the person is repentant or not, and if fact, it is He who gives the grace of repentance to that person. Hence, we are not able to judge whether the person has received the grace of final perseverance or not, whatever the external appearances are.
(Naturally, that does not take away our objective obligation to confess all our mortal sins.)
“1035 The teaching of the Church affirms the existence of hell and its eternity. Immediately after death the souls of those who die in a state of mortal sin descend into hell…”
For a Catholic to go to hell - he who has been baptized into Christ -given true life in him –
-“a willful turning away from God (a mortal sin) is necessary, and persistence in it until the end” (Catechism 1037).
One can have the grace of perfect contrition (which includes the intent to confess as soon as possible) - and be restored to life prior to any confession. (and such contrition need not any “emotion”) scborromeo.org/ccc/p2s2c2a4.htm#1452
God can reach a person in even the very last moment in ways we do not know or guess. Jesus can reach them and they can enter into life. (though we are not to presume on such in our case -but seek to prefect contrition and also seek to quickly go to confession).
I agree that it’s fairly clear that dying in unrepentant mortal sin means, pretty much by definition, that the person is going to hell.
However, I would always hold out hope that repentance takes place. Even if the person was on their death bed saying how they were not sorry for committing grave sin X, Y, and Z, I would hold out hope that, in some way known to God, the person really did repent in that final moment wherein no words are said.
If such a thing did happen, though, the person would not technically be dying with unrepentant mortal sin because they would have repented before death, even if only by a microsecond.
But to argue that someone can be saved of unrepentant mortal sin after death – even if only by one microsecond – would undermine the Church’s teaching on mortal sin and salvation. Because if they could be saved one microsecond after death when they would otherwise go to hell, why not two microseconds? Or ten? Or a day? Or a year? Or any moment until the end of time? Death would cease to be the definitive moment wherein our choice for or against God is permanently made. And any other moment we chose would be utterly arbitrary.