How does Romans 2:6-16 jive with being saved by faith alone?
This is a good example of what can happen when we take verses out of context. Romans 2:6-16 is part of a passage on God’s righteous judgment. Paul’s purpose was to show beyond a doubt the hopeless condition of natural man. The passage goes all the way to Romans 3:20.
Then, beginning in Romans 3:21, Paul introduced the righteousness that doesn’t come from works or obedience to the Law but from God, a righteousness imputed to believers by faith in Jesus Christ, that saves us from His judgment. His point in this is to show that there is no other way to be saved but by faith alone. So if you take the whole passage in its proper context Romans 2:6-16 is consistent with salvation by faith alone.