You did it. Admit it.
You did something stupid...or not, but she thinks you did, so it was stupid. You might think it wasn't worth notice, or it was stupid, or that you just weren't thinking, so there's no way anyone should hold it against you.
You don't want to talk about it. You want to brush her off. You wish she would see your point of view, which is: this doesn't matter, so stop acting like you're hurt.
Except when she tells you that you hurt her, you don't get to decide that you didn't.
That isn't how emotions work. You can't just turn them off. So whatever you did hurt her feelings. How you feel about that thing: that it was dumb, that it doesn't matter, that it was your right to do is irrelevant.
That's because you can't control emotions. You can't tell her that she has no right to feel the way she does. Because she has every right to whatever feeling under the sun she needs to have, and that's the end of the discussion.
Maybe her feelings are irrational. That doesn't mean they're invalid; it just means they don't make logical sense. Maybe she shouldn't feel hurt. That doesn't really matter; she's hurt. She says she's hurt and that your actions caused it. You don't get to decide that they didn't. This isn't because she's upset with her mom or on her period or some other nonsense. You hurt her feelings. Own up to it like a man, and ask what you can do to fix it.
You can't dictate the feelings of other people. You can't try to invalidate their feelings because that makes you a dick. So the only proper response to a woman whose feelings were hurt, she says, by you, is, “I'm sorry your feelings are hurt. What can I do to make it better?”
When you say you're sorry, you take responsibility for your actions. That's important. You're the one who hurt her and you need to own up to that, no matter how stupid it may seem to you. When you own up to your actions, you act like a man. You admit to hurting her. That takes some real guts. It's important that a man be able to take his woman's hurt in his hands and admit to causing it.
Then you acknowledge that, yes, her feelings were hurt. No matter how dumb you think this situation is, no matter how much you want to tell her to suck it up, buttercup, you admit that her feelings were hurt.
Then you ask what you can do to fix the problem. You've admitted there's a problem, you've admitted it's your fault and now, like a real man, you ask how you can fix it. She's telling you that you hurt her feelings because she wants you to fix it. She has faith that you will fix it. Your job is to ask what shape that fixing will take.
An apology is a start, which you've done. A hug might be next. She may need you to talk out how to make this never happen again.
And she might need to tell you at length how she's hurt. When she does that, she doesn't want you to argue with her. She doesn't want you to try to fix her feelings, though she may want you to try to fix the situation.
What she needs? For you to nod your head, affirm her feelings, and be supportive. You don't get to tell her she's wrong. You don't get to tell her that her feelings are wrong. You hurt them. Deal with it. And you'll get through it.