Just about everywhere in the US, people can be fired without any reason at all. It is up to someone who is fired to show that they were fired for a reason that is illegal.
Came here to say this. The only reason this French court case was a case at all is that the employer had to give a clear reason why the employee was fired. For at-will employment in America, you could also be fired for not fitting into company social culture (or for any number of other random reasons) but they don’t have to give a reason at all.
EU and American labor laws are very very different. This has been brought into stark relief by the recent goings-on at twitter, in which several European employees were certainly fired illegally.
It seems some states, like NY and California, have some sort of civil protections, however weak, that at least make companies write up justifications, document stuff, and do a PIP. It seems more rare to suddenly disappear someone unless they've done something terrible. Granted one month and 'here is an employee performance improvement plan that they will never achieve success for' is a bit dicey ethically.
While this is legally true, in practicality all companies will give a reason for being fired in the US precisely because, if the employee does bring a suit alleging discrimination, "just because" isn't a reason that's likely to pass muster with a jury.