I'd like to try a small "frame challenge" with you. Try to help me answer this imaginary question:
How can I tell my friend I was wrong to over-react when they forgot my birthday?
I have a friend who has been struggling a lot with mental health over the last few weeks and was nearly hospitalised on multiple occasions. A few days before my birthday, they nearly killed themself. Not surprisingly, they didn't wish me Happy Birthday - I'm sure they completely forgot about my birthday. The next day they contacted me and apologised for forgetting, even offered to take me out for lunch but I just said “you’re a day too late”. They keep texting me, but I haven't replied.
I'm sure you could tell that asker how to fix things: just answer some of the texts, arrange to get together, ask how the mental health challenges are going, that sort of thing. That's what should be happening here. Your friend is centering herself and her birthday as more important than what you are going through. And here you are feeling bad about it and wanting to fix things. She needs to fix things.
It's possible she can't. The world has its share of shallow and unempathetic people. Also, the word "nearly" covers a lot of ground. To me, "nearly hospitalized" means you were talking to a professional who was trying to decide whether to admit you or not. It's possible you mean you were home feeling terrible and wondering if this meant you were going to end up hospitalized. I say this just to include the possibility that she does not really know how bad things are for you.
Anyway, what to do about it? When I was going through very bad times, some friends were there for me and some friends were not. They were more irritated by the inconvenience to them of the things I was going through. As part of surviving those bad times and rising up as my better self, I learned that I don't need to take care of those people. I need to take care of me. If you want to celebrate your friend's birthday, but she feels that invitations after the birthday are not acceptable, then ok, birthday is over, move on.
Contact her the ways you usually do (texts, whatever) but don't meta talk like "are we ok? have you forgiven me? I am so sorry I forgot" and so on. Invite her to something. Share a joke or funny picture with her. Do the friend things you did before. Perhaps she will either return to your old ways of interacting, in which case: problem solved. Perhaps she will not. But that will be on her. You had a very good reason for being one day late, and if she can't handle that, she is not the kind of friend you need right now. Right?