Terminated due to complaints by harrassing coworker

What is the name of your state (only U.S. law)? Colorado
We live in Colorado and My husband has been fired from a business that works with disabled adults.
The reason - He has had two complaints filed by the same employee. An employee that has only been there a few months and has been taking notes on EVERYTHING she sees ANY staff member do wrong (except herself of course). My husband has not had a single complaint filed against him until she arrived.

The first complaint was dismissed because there was no evidence to back up her claim that he hit a disabled adult.
The second - he admitted that he did use a swear word in a sentence. ONCE.
He was dismissed for having TWO complaints against him.

Within the last two weeks (AFTER THE FIRST COMPLAINT WAS DISMISSED) he had his 4 year review, which was GLOWING PRAISE AND JOB WELL DONE.

Can a business, that runs off of State money, dismiss an employee for something that they themselves dismissed as a bogus claim?
Can we go after the employee for harassment?

Barring a bona fide, legally binding contract or CBA that expressly says otherwise, he can legally be fired for any reason that does not specifically violate the law. (This does not.)

And no, you cannot sue the state for this.