In Justice Roberts's ruling in Students for Fair Admissions v. Harvard College overturning affirmative action in the United States, he holds that "Harvard’s and UNC’s admissions programs violate the Equal Protection Clause of the Fourteenth Amendment."
Later, he writes (all emphases added):
The conclusion reached by the Brown Court was unmistakably clear: the right to a public education 'must be made available to all on equal terms'."
Brown’s “fundamental principle that racial discrimination in public education is unconstitutional.”
For the reasons provided above, the Harvard and UNC admissions programs cannot be reconciled with the guarantees of the Equal Protection Clause.
Why would the Equal Protection Clause apply to private universities? The EPC restricts certain state actions. And other Constitutional amendments do not apply to private universities.
The only opinion in this decision I could find that addressed this issue at all was Justice Gorsuch's concurrence in section II.B:
Title VI says: “No person in the United States shall, on the ground of race, color, or national origin, be excluded from participation in, be denied the benefits of, or be subjected to discrimination under any program or activity receiving Federal financial assistance.” The Equal Protection Clause reads: “No State shall . . . deny to any person within its jurisdiction the equal protection of the laws.” ...
The Equal Protection Clause operates on States. It does not purport to regulate the conduct of private parties. By contrast, Title VI applies to recipients of federal funds—covering not just many state actors, but many private actors too. In this way, Title VI reaches entities and organizations that the Equal Protection Clause does not. ... Title VI bears independent force beyond the Equal Protection Clause.
While Gorsuch is not explicitly clear on this point, I read this as saying that he believes that Harvard's policies are illegal, but under Title VI of the Civil Rights Act (42 U.S. Code § 2000d), not under the Equal Protection Clause of the Fourteenth Amendment. But this seems like a pretty major departure from Roberts' position (which I don't understand) that the EPC applies to both Harvard and UNC.