As an at-will employee (which you are unless you have a contract), you can be fired for no reason, any reason, even a bad or false reason. The only exception is that you cannot be fired for an illegal reason. Illegal reasons to terminate an employee include discrimination based on race, color, national origin, gender, religion, age, and disability. It is also illegal to fire someone in retaliation for a legally protected act, such as taking FMLA time or whistle blowing.
In your case, you were fired for something that you didn't do. That is not illegal, though it is unfair. As such, your only remedy is unemployment benefits. You cannot sue (at least successfully) your former employer for firing you for a false reason. Those who are fired for no reason or a bad or false reason are eligible for unemployment benefits, that is the remedy provided in the at-will employment scheme.
I am sorry, I know this is not what you hoped to hear, but this is the reality of your situation. My policy here is to just give the truth, rather than give you false good news in hopes of earning a positive rating from you. Hopefully, you can appreciate this.