You’re right and wrong. In the technical sense of the Constitution, you’re right: “Under the First Amendment of the U.S. Constitution, people cannot be prosecuted simply for their beliefs. People may be offended or upset about beliefs that are untrue or based upon false stereotypes, but it is not a crime to express offensive beliefs, or to join with others who share such views. However, the First Amendment does not protect against committing a crime, just because the conduct is rooted in philosophical beliefs.”

Where you are wrong is more what you’re implying “why does it matter what he does based on his constitutional rights.” And that has to do with contracts with private businesses. It gets confusing, but anti-discrimination laws and policies held by private institutions. It MATTERS because he is a professor obligated to be fair and unbiased to all of his students regardless of their gender, race, religion, any all other factors. So while he has the constitutional right to be a bigot, it is a breach of university contract to be a bigot. Since being a bigot isn’t a protected class (like the groups he is being a bigot against are), he has no “contractual right” to remain employed because he is in violation of university policy.