Trump threatens to 'close' down social media platforms

Can this even be considered a free speech issue? They aren't deleting his tweet, only displaying it alongside a fact check. Of course you can try to call into question the impartiality of the fact check but that is a long way from not deciding not to show the content.

And if he decided to "close" Twitter, it would actually be a clear case of censorship from the government and a violation of free speech.

Twitter is merely labelling a tweet as being factually incorrect, it's not hiding the content.

Well, it wouldn't be, because he _couldn't do it_. He's not a dictator, despite apparent aspirations.

Real question: As the head of the Federal executive branch, who would stop him?

Judges in every district, appeals court, and the SCOTUS. An injunction would be put in place not more than an hour after such a deceleration by the POTUS was made.