It is never really a clear cut issue on when sorry is enough. It depends on how near to each other you where or are, what the transgression was and simply if you believe the person. If someone was very rude to me that I knew to be usually an okay person and then came to me later to say sorry and that they where having a really bad day and shouldn't have allowed themselves to act out against me. I would see that as a genuine attempt to say sorry and depending on how ride they had been I would take the apology and move on or be at least open to forgiving them based on how the continue to behave.
But sorry is not a magic word, it is simply a statement or how you now feel about your actions and possibly your intentions to fix what damage you have caused. If say you damaged something of a friends by misusing it or carelessness and don't offer to either fix or pay for it then you count expect a single word to be a reasonable replacement.
As to someone making an offensive racist statement, it would depend on context. Where they trying to be funny? Fine day sorry and don't do it again. Was it a genuine statement of belief? If it wasn't a person I knew very well then I would take it as a red flag to stay away from them, even with a sorry. If I did know therm and it seemed an out of place statement then I might take it that they had just need idiotic and where indeed sorry then I could let it go, as long as they didn't continue with it.
Basically context is everything, but even with the best context it is not a magic word.