It is correct that the Church states that there is to be no new public revelation since the death of the last apostle:
Throughout the ages, there have been so-called “private” revelations, some of which have been recognized by the authority of the Church. They do not belong, however, to the deposit of faith. It is not their role to improve or complete Christ’s definitive revelation, but to help live more fully by it in a certain period of history. Guided by the magisterium of the Church, the sensus fidelium knows how to discern and welcome in these revelations whatever constitutes an authentic call of Christ or his saints to the Church.
Christian faith cannot accept “revelations” that claim to surpass or correct the revelation of which Christ is the fulfillment, as is the case in certain non-Christian religions and also in certain recent sects which base themselves on such “revelations” (CCC 67)
[quote=YinYangMom]It is my understanding the Qur’an is the direct written word of Allah as revealed to Muhammad. Through the Qur’an (or that revelation) Islam claims to be a faith stemming from Abraham, worshipping the one true God, Allah, but their Allah is not trinitarian.
Muslims believe that the Qur’an is divine revelation; the Catholic Church does not share that belief. Islam claims that it descends from Abraham; the Church recognizes that Islam makes such a claim:
The Church regards with esteem also the Muslims. They adore the one God, living and subsisting in himself; merciful and all-powerful, the Creator of heaven and earth, who has spoken to men; they take pains to submit wholeheartedly to even his inscrutable decrees, just as Abraham, with whom the faith of Islam takes pleasure in linking itself, submitted to God (Nostra Aetate 3, emphasis added, source).
[quote=YinYangMom]Why does the Catholic Church regard anything that comes from the Qur’an as legitimate since it was “revealed and written” well after the Bible was ruled upon? Doesn’t that contradict their declaration?
Non-inspired books can contain true passages, and, depending on the book, can even be true. For example, the Yellow Pages can be accurate in every detail without having been inspired by God. With non-Catholic and non-Christian religions, the Church recognizes that they can contain some truth, but insists that the fullness of truth subsists only in the Catholic Church:
[quote]First, the Church is catholic * because Christ is present in her. “Where there is Christ Jesus, there is the Catholic Church.” In her subsists the fullness of Christ’s body united with its head; this implies that she receives from him “the fullness of the means of salvation” which he has willed: correct and complete confession of faith, full sacramental life, and ordained ministry in apostolic succession. The Church was, in this fundamental sense, catholic on the day of Pentecost and will always be so until the day of the Parousia * (**CCC 830****
The Catholic Church recognizes in other religions that search, among shadows and images, for the God who is unknown yet near since he gives life and breath and all things and wants all men to be saved. Thus, the Church considers all goodness and truth found in these religions as “a preparation for the gospel and given by him who enlightens all men that they may at length have life” (CCC 843
The Catechism on Islam by Jimmy Akin
Muslims Worship the One True God by Fr. Brian W. Harrison, O.S.