Well, not exactly. It depends on the basis for the ruling, but if it’s decided on 14th amendment Equal Protection grounds, then technically that applies only to states and their subdivisions, including state universities (and I suppose the service academies, by “reverse incorporation”). The 14th Amendment says “. . . nor shall any State . . . deny to any person within its jurisdiction the equal protection of the laws.” It doesn’t say “any private institution,” it says “any State.” There’s a “state action” requirement. As a constitutional matter, private institutions aren’t prohibited from discriminating because the 14th amendment doesn’t apply to them…
That’s why for private institutions you need to look at statutory law, like Title VI of the Civil Rights Act of 1964 which prohibits discrimination by “any program or activity receiving Federal financial assistance.” So if the Supreme Court were to broadly declare any consideration of race in college admissions to violate Equal Protection, that ruling would apply directly to state universities, but not to private colleges and universities. Then the argument would be that as a matter of statutory law (Title VI), the private institutions are not allowed to discriminate because they’re receiving federal funds. There is no private right of action under Title VI to enforce this against private universities; you can’t sue HYPS, etc. on this basis. But an adversely affected private party could potentially get judicial review of a decision by the Department of Education to award federal funds to one or more private colleges and universities that were engaging in admissions practices that the Supreme Court had declared to be discriminatory under the 14th Amendment. You’d have to sue the Department of Education (or some other federal agency), and the remedy would be to vacate the award of federal funding to the offending institutions. But if Harvard were to decide it really doesn’t need federal money, it wouldn’t be unlawful as a matter of federal law for it to continue to use race-based admissions. And this might not be so implausible. Only 15% or so of Harvard’s undergrads receive federal Pell grants, worth probably around $6K or so on average. That’s a drop in the bucket to Harvard with its $37 billion endowment. As are federal student loans; according to College Scorecard, only 3% of Harvard’s undergrads take out federal loans, and those that do graduate with a “typical” debt of $6,000. Again, small potatoes. The $600 million or so Harvard receives annually in federal research grants and contracts is another matter; that’s a considerable sum of money. But there’s at least a plausible argument that Harvard’s research is a separate and distinct “program or activity” from its undergraduate education “program or activity.” Title VI prohibits discrimination in federally funded "programs or activities; thus if a single institution engages in several distinct programs and activities and only some of them are held to be discriminatory, then arguably only the discriminatory ones need to be cut off from federal funding. That would be a pretty bold gamble on Harvard’s part. But hey, who knows?