I'm sure there is some sort of state action--maybe a state subsidy, or something. Otherwise, there is no issue.
here is the decision:
http://www.ksbe.edu/pdf/doe_decision_20061205.pdf
it appears that courts have ruled that "tuition" is a contract between two entities (student, school) and that racial discrimination is prohibited in contracts. hence, private schools that charge tuition, cannot have racial preference. part of the dissent is based on this; that if the school were tuition free, it would be fine. this is a weak argument in my opinion - that tuition implies a contract exists.
Unfortunately that is untrue - Bob Jones University vs. U.S. I believe. A private school does not have the right to reject diversity or freely associate only with those that it chooses. I think Rehnquist was the only Supreme that dissented.
The issue is that in the early 1980s, the US Supreme Court ruled that private schools that practice racial discrimination can lose their federal tax exempt status. They would have to pay income taxes (which would about destroy most private schools. Ref: Bob Jones University v. United States, 1983)
The school was founded by the last queen of Hawaii as a desperate measure to preserve Hawaiian culture and ethnic identity. This is a unique situation. IMO the court did the right thing to rule in the school's favor. Native Hawaiians have a right to their own culture and language in a *private* school set up specifically to pass these traditions on.