If Monday comes and nothing happens, then what?
OP is in high school. There's a 99% chance that most at his high school already know about it, and probably will by the time Monday rolls around. And there's also a 99% chance of it already being on social media somewhere. The attention is already there. OP is receiving threats from some random guy, it won't be too long until the whole school is on his head. Getting a lawyer involved won't keep it quiet, if anything it will make it worse. Do you remember high school? Word travels faster than The Flash, and people love spreading gossip. Without getting on it, it can get bad and irreversible. At least putting something up on social media allows OP the chance to a) defend himself and b) a chance to have others come to his defense. Getting a lawyer involved also costs money and can make it even worse for him and her.
Social media can be a good tool. People use it all the time to bring other's dirt up from the past. Why not use it to show what a good person you are?
Grandstanding isn't going to help his case, the only way people are going to stop believing this rumour is for the accuser to take it back, apologise and admit she lied. There's no point in taking it to social media before Monday in case it's already cleared up by then, which, in that case, good, move on, no need to talk about it anymore. If Monday comes and goes and nothing has happened, we've already discussed further options for the OP to take.
Yes, the attention is already there, and we're trying to minimize the attention it's getting, not exacerbate it. OP has already explained that he himself understands why people will always believe the accuser first, there's going to be people who don't know about it, sees OP saying "I'm a good guy, I'm not a rapist, don't believe her" and immediately take the accuser's side. Which might lead to even more death threats.
OP is already doing the right thing and remaining (seemingly) calm about it all. Don't take it to press or social media when it could all be resolves in a matter of days.