That seems correct to me as well, but is it really?
I think they're making an argument related to misinformation / lying about a crime, which apparently is against the terms of ABC's broadcast license. I'm evidently not a lawyer.
They threatened local affiliates' licenses if they didn't pull the show, which isn't making a formal complaint against ABC for misinformation. I'm obviously not a lawyer but nothing about the FCC making informal threats to coerce self censorship seems remotely legal, beyond the fact Kimmel obviously didn't lie about anything
> surely his private employer have a right to fire him for it?
Short answer: depends on his contract.
Longer answer: if ABC fired him because of illegal threats from Carr, one could construct the argument that ABC and Carr conspired illegally to subvert Kimmel’s First Amendment rights. (Whether this is legal nonsense is beyond me.)
That would be wrong, but I think the conservative backlash was such that he'd have been very much fired anyway, FCC or no. People were publishing lists of his advertisers on X to organize a boycott.
He has a right to speak his mind, not to have a show.