"Should" there have been an investigation? Well, yes, it would be a good thing if there had been.
Does the law requires that there be an investigation? No.
can any person complain, no matter the situation Yes. I know of no laws that prohibit a person from complaining.
and cause someone to be fired? This is entirely up to the employer. NO law anywhere in the US gives a complaining employee the right to compel an employer to fire someone else. There are NO laws anywhere in the US that compel an employer to fire an employee under any circumstances. (There are some situations where the employer will face some liability if they don't, but none at all where the law says, If x happens you must fire the employee.)
However.
Assuming you are in the US (we ask for a reason) and assuming that you are not in the state of Montana, at-will employment exists. In the absence of a legally binding and enforceable contract to the contrary, you are able to quit your job at any time and for any reason, and an employer is able to terminate your employment at any time and for any reason not specifically prohibited by law. No law prohibits an employer from terming an employee because there's been a complaint about him/her.
Best practice? No.
Fair? Maybe, maybe not.
Legal? Yes.
