No, this isn't a criminal thing...like you aren't going to jail for it as it would be protected under free speech...however YES, you could be fired for it and the company sued for monetary damages civilly.
In fact, in almost all of these situations, the company's HR/legal department doesn't care who is right or wrong, they care about who could cost them the most money if they got sued.
So this is why even if they are unsure or things are murky, they will fire someone over it because they don't want to get sued and have to defend themselves in court since they have no idea what the jury is going to do and it's far cheaper for them to prevent this and hire someone else. It also gives them precedent if someone ELSE were to sue them where they can then say "Well, look at our track record from before, we took action against the employee when it was brought to our attention". This type of stuff has BIG legal implications usually because it shows intent by the company with how they acted previously and they would be much more likely to get the benefit of the doubt in the future if things are 50/50 than they would if they just let it go.
Well, I guess as someone who knows at least 3 people fired for sexual harassment at various jobs, I would have to disagree with you. Pretty sure most people from the US know about someone else who was as well.
Again, the HR department could actually believe you, but if they feel that keeping you will open them up to possible legal action which will cost them money, you could still get fired to "keep the peace" so to speak.