What to do if someone refuses to accept your apology; You said you're sorry, now what?

Don't gravel. If they want nothing to do with you after you have owned up to your mistake, then the problem is theirs. Or, it also could be that they need a little time and space. If that is the case, then give it to them. The worst thing you can do is to constantly approach them or try to contact them. Once you have said you are sorry, it is done. You did what you could. The ball is in his/her court. It could be that they eventually come around, and just need some time and space.