I don't agree with that at all. Opening an investigation after a complaint is reasonable. If you find the complaint to not be relevant, the right response is to fire the employee who made the complaint? That doesn't seem useful, it will just up the cost of complaints for employees by a lot, so you get fewer complaints and have fewer chances to fix issues.

Firing someone who has become oppositional, doesn't seem to have any interest in "patching things up", discloses confidential information and trash talks you on Twitter? That seems perfectly reasonable. There's no point in keeping them on the pay roll.