What should you do when someone won't accept your apology?

Answer 1

Nothing - if the apology was sincere. If it wasn't sincere, wait until you are actually sorry for what you have done, and apologize again, properly. If it was sincere then there is nothing else you can do, the problem is now the other person's.

Answer 2

If you apologized, and told the person you recognized your mistake, the rest is for the other person to do. You can't force someone to accept your apology, so leave it at that. You did what you had you had to do. Leave the rest for the other person to do.