The network can do whatever the hell they want. In this case they chose to remove a loser show from the lineup.
I doubt the FCC Chair had ANY real impact on their decision.
The right of freedom of association, also a 1st amendment right, says companies are free to associate or not associate with anyone of their choosing. So I would not argue that companies cannot fire employees who do not represent well the companies they work for. In fact, I recall employees fired from their jobs years ago because they flew confederate flags at their homes. So yes, Kimmel's employers are free to decide that Kimmel is not a suitable representative for their company.
I actually think, based on comments I read by Bob Iger and other execs, that this was really just the last straw for them regarding Kimmel. There is muc more to this than the comment he made about Charlie Kirk's murder.
But, according to reports I have read, the FCC chair made a comment "we can do this the easy way or the hard way", which is IMO a direct threat to Kimmel's employers. Whether they acted on it or not is irrelevant. The government should stay out of these things because the government is there to protect free speech.