On Monday, Mr. Carlson had a conversation with the Fox Corporation executive chairman, Lachlan Murdoch, to discuss a possible exit from the company, according to a person with knowledge of the meeting.

Mr. Carlson’s remarks on Tuesday, posted on Twitter — a platform run by Elon Musk, a provocateur in a similar mold as the combative, contrarian host — consisted of a three-minute monologue delivered directly to the camera. The video could violate the terms of his contract with Fox, which prevent Mr. Carlson from hosting a show on an alternate network.

One way Fox could try to prevent Mr. Carlson from posting new videos would be to seek an injunction, said Andy Lee, an entertainment lawyer with Foley & Lardner. But the network would have to persuade a judge that Mr. Carlson was doing irreparable harm to the network, such as by damaging its reputation or disclosing sensitive information. It would also have to show a likelihood that it would win at trial, another hurdle.

Mr. Carlson would probably resist all those arguments and also argue that his Twitter videos were protected under the First Amendment.

“The burden is high for this remedy, but people get injunctions all the time,” Mr. Lee said.

Bryan Freedman, Tucker Carlson’s lawyer, did not respond to requests for comment.