Realistically, formal higher education is not simply a private matter. It's a part of the complex web of accreditation, government subsidies and entrenched social institutions (not necessarily state institutions).

A university is a public accommodation. You can certainly create a book club among your friends and forbid people of the opposite gender to join or require everyone to be of a different gender, maybe you can even call it a university; but that wouldn't be the same as doing such thing on the level of a large educational facility that e.g. provides the degree of Juris Doctor that allows you to take a bar exam.

So to answer your question, "Surely you agree that the State should not...", I would say "it depends on the particulars".