"Constitutionally bound" is not the only form of "legally required." To give the example of race discrimination in admission, the Supreme Court has HELD certain public university practices to be illegal under the fourteenth amendment, but it has also opined that the same practices are illegal for private universities under the federal statutes related to nondiscrimination by entities that receive federal funding. Just about all universities in the United States receive federal funding through channels that trigger federal statutory regulation. And, as the submitted article itself pointed out, any state legislature can regulate a private university in its territory (and the example of California regulating Stanford, a private university, was prominent in the article).