It has no such authority per the Commissioner:
Federal Communications Commission (FCC) Commissioner Anna Gomez said Thursday that the independent regulatory agency does not “have the authority” to retaliate against broadcasters over what they choo...
www.yahoo.com
ABC fired Kimmel, and I believe they took into consideration many other factors besides his actual "free speech". Based on that article I think a huge reason why he was fired was trying to tell his superiors how they should do their jobs- in short, ego and insubordination. Do any of you tell your superiors or management how they should be doing their jobs?
Did you read the article? The only thing close to Kimmel telling his superiors how to do their job is this -
"Kimmel planned to address Carr’s remarks during the episode Wednesday.
He spoke with Dana Walden, co-chair of Disney Entertainment, about it before the show, The Wall Street Journal reported Thursday evening, citing people familiar with the matter. Walden and other senior executives thought Kimmel’s approach could make the case worse, the Journal reported."
That seems pretty standard. It's actually pretty responsible and respectful of him to discuss his plans with the bosses. I don't tell my bosses how to do their job but they actually do expect me to come up with ways we could be doing our job better/more efficient.
The rest of the article doesn't really agree with your analysis either. The FCC Commissioner said they don't have the authority to remove the show, yet they made the threat anyways. And have legalities stopped Trump's team before? He doesn't have the authority to send the National Guard into states to do policing.
Posse Comitatus Act - "The Act does not prevent the Army National Guard or the Air National Guard under state authority from acting in a law enforcement capacity within its home state or in an adjacent state if invited by that state's governor."