No, he shouldn't even if he could. We have the First Amendment, and that Freedom of Speech guarantees the right to say pretty much whatever you want without the government's interference. As long as it's not "criticism" that also actively promotes and incites harming or even the deaths of Israelis, Jews, or Zionists, there is nothing the government can do. Or should, speaking from the ethical standpoint that a government should respect the First Amendment. That applies to the POTUS and thus, to President Trump as well.
The internet has allowed many people to see many things, true and false. It doesn't beat actually going to Israel and spending tremendous amounts of time in the nation, but a lot of people can't do that so they may fall victim to the filth that is posted in bulk about the nation. There are a lot of hateful people who despise Israel and con others into feeling the same. The good and honest people of the world would actually appreciate Israel if they ever went to it, and see what its really all about. They'd see that "horrible racist country" isn't the best descriptor by a long shot, and I think those who actually study its history will see it's not violence, land theft, and ethnic cleansing that make the nation. Well, not in the way you suggest: Violence, land theft, and ethnic cleansing against the Jews did cause a lot of Jews to flee to Israel and help build it up to what it is today. For most people, the worst I'd anticipate them feeling is that "it's complicated", which it is in terms of the conflict; but, not hateful at all towards Israel and the Israelis.
I know many well-educated Americans (being one myself, if you excuse me tooting my own horn) and honestly: No, they're not defined by this need to criticize Israel. A few who feel the need to virtue signal to the far-left might, but then again they're often among the ones whose "well-educated" status is debated. Most people who are well-educated tend to not focus so strongly on being negative towards Israel.
The "problem" is that Trump isn't controlled by anyone, let alone Zionists. At times, many may argue it'll be better for someone to rein him in. However, even if Trump were ever so inclined to cut aid to Israel completely, he's not the only factor in determining that. Besides, even the introduction of lies spread on the internet hasn't really destroyed Israel's reputation in the States, because the truth about it, which gains people's support, is also available as well. Israel has no worries concerning America's Freedom of Speech, so even from this pseudo-"Zionist" stance you create (which is by no means reflective of actual Zionist stances), there is no reason why Trump should make criticism illegal.
I'll reiterate that Trump shouldn't make criticism illegal, for more reasons than that's illegal in and of itself. The First Amendment protects it, as it should. I don't think that any nation - Israel and the USA as well - is above criticism, nor do I want that ability to criticize without fear to be taken away. I think it's dangerous for a nation to be the arbiter of "truth" like that, which is why I'd support your ability to demonize Israel as well. It's the whole "I don't like what you say, but I'll defend your right to say it" stance. Of course, I'll still use my freedoms to criticize the #$%^ out of the sh*t you say, and hold a position that you should shut up on moral grounds; but, I'd force nothing on you.