A question I get a lot is "My boss is firing me for ___. Can he do this?"
Most employees in the U.S. are at-will, meaning that you can be fired for any reason at all, as long as it's not based on illegal discrimination (i.e., firing you because of your race, gender, religion, or other protected class). There are two exceptions this: (1) if you have a contract, or (2) if your company has an employee manual that commits to always using specific disciplinary procedures before firing someone -- if it does, it's generally obligated to follow those procedures first.
But aside from that, it's generally legal to fire someone for any reason, even if the reason is unfair or illogical.
Is it smart to fire someone for silly reasons? No, of course not. But that's a different question.