Does the President of the United States have the power to cancel a television show he deems critical of him?
It'd pretty much be in direct opposition to the concept of free speech in this country which specifically allows for public criticism of the government without retaliation.
Freedom of speech is the right to articulate one's opinions and ideas without fear of government retaliation or censorship, or societal sanction
user4012's answer does point out ways that a show could potentially be cancelled indirectly but if a president were to demand it directly, then no, they would likely be unable to do so.
That depends entirely on context. There are some narrow circumstances where the answer is "yes", but in vast majority of situations, thanks to 1st Amendment proections, the answer would be "no"
Is the TV show produced by the executive branch of the federal government?
If so, likely yes (extremely unlikely situation - to the best of my knowledge the government doesn't produce any TV shows in current pratice).
Is the TV show in violation of actual laws?
If so, he can get federal law enforcement to take actions which would cause the show to get canceled (via judicial decree, if found in violation; or simply by the producers wishing to avoid such outcome). To succeed, he would depend on said judicial decree to be positive, which is out of his hands.
Is the show in violation of FCC rules AND on a broadcast network?
If so, he can get the show fined. Presumably, he can change the makeup of FCC and have the friendly FCC make the fine too big, so the show's broadcast network would choose to cancel it.
Note that this isn't a comprehensive solution or a likely one:
FCC fines only pertain to broadcast networks. All the show needs to do is to move to cable and off broadcast, if it was on broadcast in the first place.
Or, the show/network can raise enough money from people opposing such a move (or the President) to cover the fine.
Or, Congress gets upset at this and passes the law limiting how much FCC fines can be or some other limitation to prevent this.
Does he have a big enough political influence with the right audience that he can use the bully pulpit to convince the show financers (advertisers, or companies financing it on cable) to pull the backing from the show?
If so, purely theoretically he can cause the show to be canceled without directly acting on the show itself, merely by convincing other people.
(I'm a bit leery about legality of this, but that's a question of law.se).
That's it. There's no other circumstances or methods that would allow the President to have a show canceled.
Additionally, I'm pretty sure that, if the President goes far enough to do this; they run a high chance of impeachment on 1st Amendment grounds. The support for freedom of speech - as a theoretical concept - has pretty broad bipartisan support, even if in practice both sides are quite happy to do things that are in clear violation of the 1st Amendment spirit ("Fairness doctrine", for example).