In the name of free association, private institutions can fire anyone they want, generally, under the at-will doctrine, with or without cause. The less clear-cut issue is when government is involved, and when does retaliation become stepping on someone’s free speech rights. From el gato malo at boriquagato.substack.com:
assessing the accusations of cancel culture
the story of the last week has been one of a worm turning and a society as one standing up to express that which has long inhabited hearts but, for reasons of fear and intimidation, has been absent from voices and actions:
we are deeply, astonishingly, utterly and bone deep fatigued with having to cohabit with crybully cadres of smug, nasty, awful people who mistake such traits for virtue and hector others relentlessly, drunk on self-entitled zeal.
and it has suddenly become OK to tell these people off again.
it has become OK to separate from them, fire them for having low character, and require that they leave the civilized company of others.
this is becoming a full blown “they’re eating the dogs, they’re eating the cats” moment where an avatar rises as a form of societal acid test confluencing with a societal permission structure and it becomes OK to become angry and it becomes OK to defend oneself from the endless punch no punchbacks predation of these “protected” classes of hallucination peddlers endlessly demanding that you call men women, the morbidly obese healthy, and anyone not exactly like these obstreperously assaultive karentopian conmen (and women, and them/thems) fascists.
you can never really guess what the avatar is going to be. it emerges unexpected from the social current and galvanizes a zeitgeist making that which could not be said at once into common currency.
and here we are.
the omerta is over and once more, speech has consequences.

(dust appears ot have more calories than i would have suspected)
Words can never hurt me as the childhood saying goes.
The 1A protects speech that you don’t like.