What are my rights as a manager who has been slandered? Be pleased that the investigation proved that the claims were groundless.
I know that there are retaliation laws, but could this employee be fired for making such claims? Yes, but your company would be on massively shakey legal ground as the employee in question will immediately file a claim with the government for wrongful termination - retaliation for making a SH complaint.
I am angry, let alone humiliated from all of this. There seems to be plenty of laws to protect those who make claims of harassment, but where are the laws to protect those who are falsely accused? I don't blame you one bit for being angry. The difficulty here is proving that the claim made against you was knowingly false - that is, the employee deliberately made up the complaint to cause you problems. That is quite a different thing than the investigation concluding that the claim was frivolous. In the absence of a "smoking gun" (e.g. your employer has a copy of an email she sent in which she stated she was going to make up a SH complaint) it's virtually impossible to prove that the employee made it all up, as opposed to her being mistaken, misinterpreting your actions, being a wack-job, etc.
Sorry about all that. I don't blame you for wanting her gone but terminating her employment or even disciplining her without positive proof puts your employer in great legal jeopardy. Even defending a groundless claim for retalation is massively expensive.