There is some Supreme Court precedent indicating that public universities must be open to sectarian religious organizations (see Rosenberger v. Rectors and Visitors of the University of Virginia, 515 U.S. 819 (1995)). But that case isn't really on-point -- it stands for the simple proposition that religious groups must be admitted to campus on an equal basis with secular groups, and that Establishment Clause concerns can't be used as a ward against this.
No, the case that most clearly comes to mind is Bob Jones University v. United States, 461 U.S. 574 (1983). There, the IRS revoked BJU's tax-exempt status due to its religiously-inspired policy against inter-racial dating. The court upheld the action because racial discrimination in education is contrary to public policy. More broadly, while the government certainly can't prohibit private groups from excluding racial minorities, it does not have to give such behavior its imprimatur. In setting public policy, and in managing its own affairs, it is well within the government's right to treat racially-inclusive groups differently from their exclusionary peers. The government is allowed to set policies that prefers equality over discrimination.
Responding to BJU's First Amendment claim, Chief Justice Burger wrote:
On occasion, this Court has found certain governmental interests so compelling as to allow even regulations prohibiting religiously based conduct. In Prince v. Massachusetts, 321 U.S. 158 (1944), for example, the Court held that neutrally cast child labor laws prohibiting sale of printed materials on public streets could be applied to prohibit children from dispensing religious literature. The Court found no constitutional infirmity in "excluding [Jehovah's Witness children] from doing there what no other children may do." Id. at 171.... Denial of tax benefits will inevitably have a substantial impact on the operation of private religious schools, but will not prevent those schools from observing their religious tenets.
The governmental interest at stake here is compelling.... [T]he Government has a fundamental, overriding interest in eradicating racial discrimination in education -- discrimination that prevailed, with official approval, for the first 165 years of this Nation's constitutional history. That governmental interest substantially outweighs whatever burden denial of tax benefits places on petitioners' exercise of their religious beliefs. The interests asserted by petitioners cannot be accommodated with that compelling governmental interest ... and no "less restrictive means," ... are available to achieve the governmental interest.
Id. at 603-04.
It is relatively uncontested, then, that in certain contexts religious practices must yield in the face of anti-discrimination norms. I can't imagine that if the CLS was asserting a religious requirement to exclude Blacks, this case would be before the Supreme Court -- it might theoretically be possible to distinguish Bob Jones, but I doubt any court would take the bait.* So as far as I see it, the question is simply whether the Court is willing to put anti-gay animus on the same constitutional level as racism. Or more accurately: whether it is willing to defer to a governmental determination making that value judgment. At bottom, Martinez doesn't present a particularly novel issue of First Amendment law. The only thing new is whether this Court, the Roberts Court, is willing to allow a government body to proclaim a public policy against homophobia. That it is considering prohibiting this is indicative of just how shallow both its "minimalism", and its protestations of non-"activism" (whatever that means), are.
* There are many theoretical difficulties in terms of how to distinguish between government properly refusing to issue its stamp of approval towards discriminatory organizations, and government improperly acting to suppress those groups entirely. If the government said "fire fighter protection is a privilege that we're withholding from racist organizations", I doubt it would be sustained. But I don't think those theoretical problems are in play here, precisely because we've already started the line-drawing process with Bob Jones, and, were this case one of racial exclusion rather than heterosexist exclusion, we'd know exactly which side of the line the CLS would be on. It's not the theory issues that make this case "hard", it's whether courts are willing to let a government body treat homophobia the same way as it does racism. And I don't see any principled reason why it should be precluded from doing so.