“... Private institutions that receive federal funding must also adhere to federal anti-discrimination laws, such as those applicable under Title IX.
There are some exceptions to this rule. Private colleges and universities that accept government funding or which otherwise engage with government closely may be required to adhere to the First Amendment more closely. State governments may also pass statutes requiring private universities to respect free speech rights as a matter of state law, even when the US Constitution imposes no such requirement. For example, California law applies First Amendment protections to both public and private universities. Congress also has the power to propose and pass federal laws which would require private universities, by statute, to adhere to various free speech guidelines.
https://campusfreespeechguide.pen.org/the-law/the-basics/
Adhering to the First Amendment applies to both public and private universities.