Trump isn't the chap to change the laws, of course.
But the notion of defamation/ reputational damage is a concept that already exists in the US as a limitation on free speech. So it is only a question of where the line is drawn.
It is kinda mythological that the first amendment is unabridged. It is abridged where harms are done to the vulnerable. eg sharing pornographic pictures of minors is a form of speech and properly a criminal offence.
If someone lies about you in a damaging way, I also see no problem with the law providing a remedy.
A workable defence for the author is that they told the truth.
Newspapers ought not to be in the business of dishonest reporting. For myself, I don't think the Washington Post and the NYT are.
Trump dislikes the truths and the opinions that are told about him and wants them to stop. As folks with dictatorial leanings tend to. He's trying to scare them into giving him a pass. And at the same time, sucking up to the folks who are trained to hate the media (ie Republicans).
Is there anyone left that he doesn't want to crush?