Only in California.
The First Amendment provides a student essentially no protection from discipline by a private university. Manhattan Cmty. Access Corp. v. Halleck, 139 S. Ct. 1921, 1928 (2019) ("The text and original meaning of those Amendments, as well as this Court's longstanding precedents, establish that the Free Speech Clause prohibits only governmental abridgment of speech. The Free Speech Clause does not prohibit private abridgment of speech."); Vaynberg v. Seton Hall Univ., No. CIV.A. 09-4999 FSH, 2010 WL 4510904, at *5 (D.N.J. Oct. 26, 2010) ("In order for the First Amendment to apply, the challenged conduct must be deemed 'state action.' Seton Hall is a private, Catholic university. ... Because there is no evidence from which a reasonable fact finder could conclude that Seton Hall's conduct relevant to this lawsuit was “state action,” Seton Hall is entitled to summary judgment.")
Some states, however, have passed laws requiring private schools to provide some of the protections of the First Amendment. The most robust of these is California's Leonard Law, which essentially requires private schools to adhere to the First Amendment. Other states also recognize some measure of free-speech rights for students at private institutions. For instance, both the Pennsylvania and New Jersey supreme courts have held that their state constitutions' free-speech clauses (which, unlike the First Amendment, say nothing about the government) protected peaceful protesters who distributed leaflets on the campuses of private colleges.