Your remedy here would be against your specific accuser, not against your employer. Your employer is free to fire you or demote you based on rumors, true or false.
As an at will employee (which you are, unless you have an employment contract or union), you can be fired for no reason or any reason, even an unfair false one. The exception is that you cannot be fired for an illegal reason. Illegal reasons to fire someone include discrimination based on race, gender, religion, age or disability. It is also illegal to fire someone or treat them differently in retaliation for a legally protected act, like whistle blowing or taking FMLA time.
In your case, you were fired based on false accusations. This is not illegal on the part of your employer. That said, you may have a defamation of character claim against the specific person who made the accusation. To show defamation, you have to show that a statement was made to 3rd party, the statement was false, and that you were damaged. All of those are present here, so you may have a defamation claim.
I am sorry, I know this is not what you hoped to hear, but this is the reality of your situation. My policy is to just give the truth, rather than give you an answer that will be pleasing now, but will only result in more frustration and disappointment for you in the end.
Please rate my answer as one of the top three faces when you are ready- I cannot control the law as it applies to your case, this rating describes whether my answer was correct (not that the answer was good news) without this I do not get credit for my time. If you are not satisfied with my response, please press "Continue Conversation" and allow me to address your issues. I only get paid credit for my time if you rate my answer positively, I answer these questions on the honor system, please do
not abuse it.