In my 4 and a half years of playing WoW, I've had a few run-ins with folks who clearly stepped over the line of common courtesy into the territory of "oh no you didn't just say that to/about me>.<"
The things said are irrelevant. Because everyone has a bad day now and then. Or says something they don't mean and immediately regret. It's what happens after you are horribly rude or offensive that matters.
The mature thing to do is to own your behavior. Take responsibility for having behaved like an ass. And simply say, "I'm sorry."
You may or may not have noticed something very important above. It's that I didn't say "I'm sorry, but..." Or "I'm sorry you were offended." That is, in fact, a very important period. If you have upset someone else, they do not want to hear your myriad justifications as to why. Your rationalizations are irrelevant. Your childhood traumas not a valid excuse.
If you want an example of how not to apologize, just look at that woman who tossed a poor friendly kitty into a recycle bin. She is a textbook example of someone "apologizing" and in fact digging themselves into a bigger hole.
Of course, in the World of Warcraft, no one is going to make you apologize if you don't want to do so. You can totally choose to piss someone off and continue on merrily as though nothing happened. But here's the whole truth of the matter: especially if your offense happened publicly, that person isn't going to forget about it.
Oh yes, I know, it's "humiliating" to apologize. It puts you in a less powerful position than the person being apologized to. Or it makes you feel bad. Tough luck! How do you think you made the other person feel? Suck it up already -- it costs you nothing but a moment of misplaced pride to do the right thing and apologize. And it can sour your relationship with someone else permanently to be too proud to admit you messed up and to ask for forgiveness.
I've been on both ends of this spectrum. I've lost my temper and been an ass. And immediately, publicly apologized to those who bore witness to it, and to the person I put in a bad position. I've also been the wronged person, who had someone else sat from raids when they refused to apologize, and as soon as they did -- through gritted teeth-- gave them a second chance. No, we didn't become BFFs, but we did in fact become more friendly than we had been before the incident of unbridled rudeness.
Hey kids! Get off my lawn! And if that ball comes into my yard, I'm keeping it!