Can Non-Christians do what is good?
The answer to the question should be straightforward,of course they can! Christ commends the faith of a pagan centurion (Matthew 8:10-12), the gratitude of a Samaritan (Luke 17:17-19), the loving devotion of a Phoenician mother for her daughter (Mark 7:24-30), the national repentance of the city of Nineveh (Matthew 12:41) and he insisted that outsiders loved their friends and families as surely as his own followers loved theirs (Matthew 5:46-47). The pre-Christian life of Cornelius was a sight to behold. (The real question is not can they be good; the real question is who enables them to be good? And the answer to that's crystal clear--God of course!)
It's a crass form of Christian imperialism that insists that the only good that's really "good" is Christian. There must be many reasons that lead us into such silly conclusions and a sinister elitism is probably one of them. We're a cut above the rest, don't you see! (That'll be the day.) But another reason for it is bad theology and bad handling of biblical texts.
Non-believers are "in the flesh" and that means (so we're told) that they can do no good, only evil. Christians are "in the Spirit" and the Spirit enables them to do what is good. So what are we supposed to do with the vast amount of good so plainly to be seen in non-Christians? We're to do what one gentleman told me we're to do--deny that it is truly "good". It's counterfeit. We stood in the street and twice read the whole of Acts 10 while he looked for some way to show Cornelius wasn't good.
Romans 7:14-25 (a discussion for another time) is used to show that a non-Christian cannot be or do good. Whoever it is that Paul is describing "delights" in God's law and inwardly is a slave to it (7:22,25) and that itself is a profound good. Galatians 5:16-26 is used for the same purpose but it is a misuse of scripture. Paul is not dealing with the question: "Can a non-Christian do good?" He is insisting that to follow and live after the Spirit will result in goodness, will result in fulfilment of the torah (law) rather than a loose moral life. The question is more like, "Will the fruit of the Christian gospel and life be immorality and wickedness?" and his answer is that it will result in the fulfilment of the torah (5:13-14,23 and compare Romans 8:4 and 13:8-10).
God not only gives food and fruitful seasons to non-believers, he gives them "everything else," grand character included (see Acts 14:15-17 and 17:25).