I think by righteousness you then mean sanctification (i.e. holiness) correct? Yes, works can help sanctify us. Remember, that our works and merits are simply Christ’s gifts returned to him. It is confusing in Protestantism that our works are somehow something separate from Christ’s work. Rather, as CCC#2009 says: “Our merits are God’s gifts.” This is also in the council of Trent:Jesus Christ Himself continually infuses his virtue into the said justified,-as the head into the members, and the vine into the branches,-and this virtue always precedes and accompanies and follows their good works, which without it could not in any wise be pleasing and meritorious before God . . . God forbid that a Christian should either trust or glory in himself, and not in the Lord, whose bounty towards all men is so great, that He will have the things which are His own gifts be their merits. (Trent 6.16)
Think of the Biblical figure of the vine and the branches (John 15) mentioned above, and how we are called to bear much fruit as part of the vine. If the branches are producing fruit, it is only because the flow of power comes to the branches from the source––the vine, Christ. If we deny that we can do good works by the power of grace, who are we denying? Ourselves? No! We deny Christ and the power he has generously shared with us as his branches!
A small sampling of other examples of how such works are beneficial to our righteousness:*Mark 4:20 But those that were sown upon the good soil are the ones who hear the Word and accept it and bear fruit…
2 Timothy 3:16-17 All Scripture is inspired by God and profitable for teaching, for reproof, for correction, and for training in righteousness, that the man of God may be complete, equipped for every good work.
Romans 2:13 For it is not the hearers of the law who are righteous before God, but the doers of the law who will be justified.
Romans 6:16 Do you not know that if you yield yourselves to any one as obedient slaves, you are slaves of the one whom you obey, either of sin, which leads to death, or of obedience, which leads to righteousness.
2 Corinthians 9:6-8, 10 …he who sows sparingly will also reap sparingly, and he who sows bountifully will also reap bountifully. Each one must do as he has made up his mind, not reluctantly or under compulsion, for God loves a cheerful giver. And God is able to provide you with every blessing in abundance, so that you may always have enough of everything and may provide in abundance for every good work. … He who supplies seed to the sower and bread for food will supply and multiply your resources and increase the harvest of your righteousness.*:o