Heh thanks for the replies so far.
I was hoping against hope for a clear “you are clearly right who would think otherwise?” but of course I should have known better 
I just can’t bring myself to accept, at least for now, that when I, as a favor, decide to help someone out with a game they’re playing, and when I make a simple mistake which loses the game, then if their response is to angrily blame me for the loss then I should apologize to them. I feel like the anger is uncalled for–I was doing them a favor!–and an apologize would incorrectly legitimize that anger.
Alright, so TMI, here’s what happened. A deep lifelong friend who is very mathephobic was playing an online game and needed to do some calculations in an argument she was having with someone else playing the game. She asked me what fifty million divided by two million is. I was thinking “fifty something” and was just doing the bit where I cancel out zeroes in my head, and while I was doing that she said “fifty million?” My train of thought interrupted, and my brain already primed to say “fifty something” anyway, I unthinkingly replied “yes.”
She took this figure to the conversation, and so of course turned out to be wrong in the argument.
She found this extremely humiliating–extreme mathephobia, like I said–and as soon as we discovered her mistake, she angrily told me “go away.” Okay, fine, she feels humiliated, I understand, and I leave.
But I believe that at this point it was her responsibility to say something conciliatory ere we spoke again. Instead, the first thing she said to me was an angry and bitter sounding “I was fucking humiliated!”
At this point, it seemed the conversation was going to become one about how I had fucked up and should feel like a bad person for it–a sentiment with which I disagreed–so I explained that I would not be participating in the conversation. And as you can imagine, things only went downhill from there.
Much later in the conversation she insisted I only had to say “I’m sorry” from the beginning and everything would have been fine.
But I argued (and still would) that since she opened with “go away” and continued later with angry and bitter blaming sentiments, and since once she said “go away” it was her responsibility to continue the conversation and set its tone, an apology from me is not just non-obligatory, it would have been downright wrong. She was wrong to hold anything against me. I can understand her feeling angry, but a responsible person would recognize that the anger is illegitimate. To apologize to someone expressing illigitimate anger, it seems to me, is not just non-obligatory but downright wrong. It is to acquiesce to a falsehood. I was not her enemy, I was her friend, and I did my best for her as a friend, and for that I refuse to apologize. 
Yeah, way TMI I know. But there it is. (Sorry, I just feel this need to spill because the conversation was just about the worst I’ve ever had with anyone, ever, with the possible exception of the one that ended with me calling my dad a “stupid fat man” and getting hit. Things got really, really bad. Next morning, I’m still extremely upset. But I don’t want to go into details about the conversation–I just want to get clear about what started it.)
I know she may not be coming off in the best light due to this story, but you have to understand the depths of her math fears. She was literally abused (in a psychological sense) by math teachers as a kid, for example. So as a friend, I understand how upset she was. But I don’t think I’m being petty by having refused to apologize. No matter how angry anyone gets, a basic level of respect is still called for especially between friends, and that means that when you are angry at someone for making an innocent mistake when they were trying to help you, you do not hold it against them, realizing instead that the problem lies within yourself. If you hold it against them, you’ve made the very idea of an apology worthless.
