Actually I don;t think that's true, and that was one of the unsuccessful arguments made on behalf of VMI, that it would "destroy single sex education".
That was rejected, and students at, i.e., Wellesley still can use govt loans and the school can still receive govt funds.
In other words, "state school" has a limited meaning, in this area, to schools that are primarily state-funded in the traditional sense.
If what you said is true, then it would be legally permissible for a private school to refuse admission to all minority applicants. I'm not sure that is legally permissible today.