4th Circuit Case probably will be overturned if S.C. takes cert. There is a case on point (somewhat). Rendell-Baker v. Kohn, 457 U.S. 830 (1982). There, the Court ruled a private school wasn't bound by the 14th amendment because there's no state action. Almost all of the private school's operating budget derived from public funds. However, that case is distinguishable a few ways. Most importantly, the case here was about employment practices, not policies concerning student conduct or admissions.
The opinion in the pending Harvard admissions case may help us get a sense of what the Court thinks in this area, but I think the o.p.'s question is open and won't be addressed there. From my perspective, it seems the current court is very critical of affirmative action. Robert's will likely write the opinion (in at least the UNC affirmative action case). Invalidating an affirmative action program in the Seattle and Louisville schools, he wrote that "the way to stop discriminating on the basis of race is to stop discriminating on the basis of race. See Community Schools v. Seattle School District.
I think on the overarching question, private institutions are generally free to create their own policies, procedures, etc. Perhaps the o.p. is right that the States preserve the right to regulate private institutions as they see fit. And, imo, it appears the Court is more sympathetic to state's power at the moment. Thus, it seems unlikely the current S.C. would challenge a State's regulations of private colleges. National Pork Producers v. Ross exemplifies this trend. Thus, a case on o.p.'s question might be a perfect storm for conservative revanchism.
Personally, I don't think this kind of law would be prudent. I think it contradicts a conservative perspective on the relationship of private actors to the state in a number of ways. And while it might be a temporary win, I'm not sure this is outcome doesn't come back to hurt conservatives in a couple of different ways.
Indeed, I think this National Pork Producers case is a good example of why these kinds of contradictions will eventually undermine conservative goals. In that case, California was given a free hand to set national commerce regulations. I think conservatives are excited about this because it will give them cover for actions by Texas and Florida (for example). I think the first big flashpoint in this area will be over social media regulations. And there, I don't think it's a good idea to give state's such diverse powers. There is a real benefit to a unified national market. Truly, I think this leads to big national conflicts which can only be solved by Federal policy making where all states are represented (even if that representation is somewhat unequal). Our federal system is a delicate, and we should be worried about upsetting the balance of power too much.
FL and TX can jointly mandate using one specific conservative sex education textbook, so that it becomes the standard.
