In my mind, if I apologize and it's sincere but the person doesn't accept it then that's on them. I'm not saying it's their fault that they don't accept the apology but sometimes there's nothing you can do about how someone feels. Also, I don't know the situation but just because you offended someone doesn't mean you were being actually offensive, though sometimes you could have been. People have hang ups so it's not always you.
The only thing that you can do is just be sincere in what you do and hope for the best but sometimes people have other hang ups that may not even involve you.