Would you sue if this happened to you?

Defamation of character.

Fuck social media.

If they are lying then yes.
Defamation is a crime.

If it's definitely not true and it's affecting your job/friends/reputation, and you have physical evidence, then talk to lawyer about slander/defamation. Can't hurt.

I guess I would have to sue them to clear my name.

Probably find her and slap her in the face.

But usually when they accuse you its because you broke their heart or something or they regret their decision to fuck u. I always try to leave on good terms with women im with.

No i would kill. Only rapists make false accusations.

Yeah, if I thought I had the case/evidence to win. Might be hard to prove.

After they get sued, they'd disappear.

never seen again.

This happened to my second brother. Some girl made false rape accusations towards him after he talked shit to her because she wouldn't fuck him (dick move on his part, not justifying that one bit) then some people came to his apartment to try & jump him but unfortunately for them they didn't know my first brother & one of his friends were there to even the odds. When I heard about it I went on a social media crusade every single day for about 3 weeks gathering as much evidence as I could & gave it to my brother & his lawyer. Still got the screenshots to prove it, her story wasn't making sense, her friend's stories didn't match hers & it turns out this wasn't the first or even second time she's lied about rape. Easily won case on our part.

I can say without question that I’d kill a person over less.

But sure, sue them first.

I hate when this happens because it makes it so horribly more difficult for woman who have truly been through that to be believed

Considering lying about something like that can ruin someone's life for something they never even did, yes. It's absolutely despicable.

People already do this to me (at least something similar) Believe me if I could, I would.
I would find out where they live and fuck them up.

Well before suing anybody, I'd at least give them a chance to take back what they said before suing them. So I'd tell them to take it back or I'll sue them.