The First Amendment is still implicated even in private disputes when one party requests that a court take some action that restricts the speech of the other pa

The First Amendment is still implicated even in private disputes when one party requests that a court take some action that restricts the speech of the other party. The court can't take the requested action (e.g. an injunction stopping publication) if it violates the first amendment. This is a bit of a simplification but the name in legal circles for the concept you are talking about is the "state action" requirement.