Romans 2 describes someone who will be accepted by God:

He will render to each one according to his works: to those who by patience in well-doing seek for glory and honor and immortality, he will give eternal life; but for those who are self-seeking and do not obey the truth, but obey unrighteousness, there will be wrath and fury. There will be tribulation and distress for every human being who does evil, the Jew first and also the Greek, but glory and honor and peace for everyone who does good, the Jew first and also the Greek. For God shows no partiality.

For all who have sinned without the law will also perish without the law, and all who have sinned under the law will be judged by the law. For it is not the hearers of the law who are righteous before God, but the doers of the law who will be justified. For when Gentiles, who do not have the law, by nature do what the law requires, they are a law to themselves, even though they do not have the law. They show that the work of the law is written on their hearts, while their conscience also bears witness, and their conflicting thoughts accuse or even excuse them on that day when, according to my gospel, God judges the secrets of men by Christ Jesus…

So, if a man who is uncircumcised keeps the precepts of the law, will not his uncircumcision be regarded as circumcision? Then he who is physically uncircumcised but keeps the law will condemn you who have the written code and circumcision but break the law. For no one is a Jew who is merely one outwardly, nor is circumcision outward and physical. But a Jew is one inwardly, and circumcision is a matter of the heart, by the Spirit, not by the letter. His praise is not from man but from God.

Now here is Acts 10 when Peter is sent to the household of Cornelius:

So Peter opened his mouth and said: “Truly I understand that God shows no partiality, but in every nation anyone who fears him and does what is right is acceptable to him.

Then James made this suggestion at the Jerusalem Council in Acts 15:

Therefore my judgment is that we should not trouble those of the Gentiles who turn to God, but should write to them to abstain from the things polluted by idols, and from sexual immorality, and from what has been strangled, and from blood.

Which was agreed to and sent out to the churches as a decree:

The brothers, both the apostles and the elders, to the brothers who are of the Gentiles in Antioch and Syria and Cilicia, greetings. Since we have heard that some persons have gone out from us and troubled you with words, unsettling your minds, although we gave them no instructions, it has seemed good to us, having come to one accord, to choose men and send them to you with our beloved Barnabas and Paul, men who have risked their lives for the sake of our Lord Jesus Christ. We have therefore sent Judas and Silas, who themselves will tell you the same things by word of mouth. For it has seemed good to the Holy Spirit and to us to lay on you no greater burden than these requirements: that you abstain from what has been sacrificed to idols, and from blood, and from what has been strangled, and from sexual immorality. If you keep yourselves from these, you will do well. Farewell.

So what shall we say to these things?  Peter learns that God shows no partiality because all Gentile god-fearers are accepted by Him.  Paul teaches that God shows no partiality because the Gentiles who persevere in godliness will be granted immortality and will condemn those who are Jews who do not obey God.  At the Jerusalem Council, likewise, the issue is what obedience is required, and the Gentiles are told to keep obeying basic morality: no vampirism, idolatry, or sexual deviancy.

So it is hard to believe that Romans 2 would be presenting a hypothetical perfect obedience that could hypothetically produce glory, honor, and immortality apart from Christ.

It might be worth considering that Romans 2 and Acts 10 and 15 are all dealing with people who know the true God.  I find it hard to believe that the Romans or the god-fearing Gentiles would ever dream that sinless perfect obedience is a possibility.  First Kings 8.46, Psalm 130.3-4, Psalm 143.2, and Ecclesiastes 7.20 are not the only passages that would rule that out from consideration and teach that God only can accept us by grace.  While we tend to read Romans 1.18ff as an argument that everyone sins, Paul does not list universal sins but the depths of apostasy.  Furthermore, while we use Romans 3.23 as asserting the same thing as the passages I just listed, Paul does not seem to be using the word “sin” to refer to any disobedience in thought, word, or deed, but for falling into real unbelief and apostasy.  After all, we all have ongoing sin by the first definition, but Paul says, “God shows his love for us in that while we were still sinners, Christ died for us” (Romans 5.8).  Whatever a “sinner” is for Paul, it is something that we are no longer.  And since it is very true that we commit ongoing sin (i.e. disobedience) and need God’s ongoing forgiveness, Paul has something more than that in mind.

On the other hand, when Paul preaches to pagans he does seem concerned to show them that they have no hope of winning favor from God by their works.  He told the Athenians,

The God who made the world and everything in it, being Lord of heaven and earth, does not live in temples made by man, nor is he served [healed?] by human hands, as though he needed anything, since he himself gives to all mankind life and breath and everything.

So the fact that we all live only by grace, not because of what we think we can do for God is important in confronting paganism.  It just doesn’t seem to be the primary issue in Acts 10, 15, or Romans 2.