Twitter is merely labelling a tweet as being factually incorrect, it's not hiding the content.
I know this is a very massive hypothetical, but it’s one I’ve wondered for a while. Basically, as the head of one of the branches, he could have subordinates forcefully removed, but who’ll forcefully remove him in this case?
In real terms, what are you imagining here? Trump having the NSA execute a DDOS against Twitter? I feel like you have to get to some pretty fantastical action-movie type plots to make this happen.
It would have to happen through the courts, and the courts won't allow it.
They can bully Twitter and threaten to e.g. withhold federal contracts (though even this runs into legal trouble) but how does the executive branch just "shut down" a platform?
You can't just send in the FBI and put a halt on things.
This will just be more of his mindless rage that a certain portion of the population gobbles up. His real goal is to discredit Twitter et al, which is unlikely to have much impact.
The good news is that the presidency (and the leader of the executive branch) is very time limited. The constitution is so clear that there is no wiggle room at all, no matter what happens between now and January next year, the only way he stays in power is by winning the election. Also it seems pretty clear there will be a democrat as acting president if we fail to vote for a new one, but that is mostly coincidences and luck this time around.
The last four years have shown that there are no real checks and balances and they depend on one party keeping its own members in line, and that the GOP have moved far, far, to the right as they are loosing the potential to win fair elections. Winding this down is not going to be pleasant, and in the long run we desperately need reforms. Also it seems like the current best case is that the GOP get voted out everywhere, but that is also a terrible outcome, we need a real opposition party and competition of ideas.
In this bizarre hypothetical, Twitter would presumably just fail over to servers outside the US, as would all other significant tech companies. Or, y'know, California might secede. It's such a weird proposition that it's hard to speculate about.
That probably won't happen though, this is really just about stirring up a frenzy of right-wingers so Twitter will have to bow down to them and give them more and more concessions in hopes that somehow, someday they'll stop accusing everything of being biased against them.