ABC is free to suspend Jimmy Kimmel's show for whatever reason it wants to (at least unless there's some contractual limit on that, which I doubt). The First Amendment doesn't constrain private companies such as ABC; indeed, it protects networks' right to choose what programs to air.
But if the government coerced ABC into suspending the show, through threats of retaliation, that would have likely violated the First Amendment. And we know that because of last year's Supreme Court decision in NRA v. Vullo.
In NRA v. Vullo, financial companies cut off or limited ties to the NRA, allegedly because they were pressured to do so by the New York state Department of Financial Services. If the companies had acted on their own, that wouldn't have violated the First Amendment, since the First Amendment doesn't generally apply to actions by private businesses. But the Court held that, if New York had coerced them, that coercion by the government would be unconstitutional. (The case came before the Court early, before there was any factfinding, so the Court left it to lower courts to resolve those factual questions.) To quote the Court:
