It depends entirely on his contract - I'd expect most CEOs to have a clause about their conduct causing harm to the organisation. Even if not, it's legal to convince someone to leave, which is what usually happens in public embarrassment cases.
I don't know where you're getting this idea from, but you really should check your sources.
There is a limited prohibition against screening for political affiliations enforced by the Federal government when filling certain civil service positions, but that's about it.
For a good summary of what types of discrimination can and cannot trigger legal liability, see here:
No it's not. Indeed, you run a legal risk in NOT firing such a person given that keeping them around (especially in a senior position) creates a demonstrably hostile work environment.
