Dear {Recipient},
I would like to apologize for {problem}. It has come to my attention that you've pretty much told everyone what a horrible person I am because of {problem}, so I really feel like I should apologize to you. I mean, it's not like{problem} was in any way my fault, but now that you've gone and blamed me for it everyone "knows" it was. So, thanks for that, and I'm really sorry you felt the need to do that.
Also, I'm sorry you're such a petty person that you couldn't just say any of these things to my face. It's really telling when someone like you can't even resolve {problem} without resorting to high school level gossip techniques. Way to act like an adult.
So, please accept my obviously sincere apology. I know you must get a lot of these, since you're always demanding people say they're sorry to you. And, of course, each of those must be as sincere as the one I'm writing to you right now.
Your humble and groveling servant,
{Sender}
