No, it's not constitutional for the University of Oklahoma to expel students for racist speech [UPDATED in light of the students' expulsion]
Some University of Oklahoma students in the Sigma Alpha Epsilon fraternity were videorecorded singing (as best I and others can tell),
There will never be a nigger at SAE
There will never be a nigger at SAE
You can hang him from a tree
But he'll never sign with me
There will never be a nigger at SAE
University of Oklahoma president David Boren said, "If I'm allowed to, these students will face suspension or expulsion." [UPDATE: The president has indeed expelled two of the students.] But he is not, I think, allowed to do that.
1. First, racist speech is constitutionally protected, just as is expression of other contemptible ideas; and universities may not discipline students based on their speech. That has been the unanimous view of courts that have considered campus speech codes and other campus speech restrictions - see here for some citations. The same, of course, is true for fraternity speech, racist or otherwise; see Iota Xi Chapter of Sigma Chi Fraternity v. George Mason University (4th Cir. 1993). (I set aside the separate question of student speech that is evaluated as part of coursework or class participation, which necessarily must be evaluated based on its content; this speech clearly doesn't qualify.)
UPDATE: The university president wrote that the students are being expelled for "your leadership role in leading a racist and exclusionary chant which has created a hostile educational environment for others." But there is no First Amendment exception for racist speech, or exclusionary speech, or - as the cases I mentioned above - for speech by university students that "has created a hostile educational environment for others."
2. Likewise, speech doesn't lose its constitutional protection just because it refers to violence - "You can hang him from a tree," "the capitalists will be the first ones up against the wall when the revolution comes," "by any means necessary" with pictures of guns, "apostates from Islam should be killed."
3. To be sure, in specific situations, such speech might fall within a First Amendment exception. One example is if it is likely to be perceived as a "true threat" of violence (e.g., saying "apostates from Islam will be killed" or "we'll hang you from a tree" to a particular person who will likely perceive it as expressing the speaker's intention to kill him); but that's not the situation here, where the speech wouldn't have been taken by any listener as a threat against him or her. Another is if it intended to solicit a criminal act, or to create a conspiracy to commit a criminal act, but, vile as the "hang him from a tree" is, neither of these exceptions are applicable here, either.
4. [UPDATE: Given the president's letter, it's clear that the students are being expelled solely for their speech, and not for the reason discussed in the following paragraphs.] Some people have suggested that the speech may be evidence of discriminatory decisionmaking by the fraternity in admitting members. A university may demand that groups to which it provides various benefits not discriminate in admissions. See Christian Legal Society v. Martinez (2010). Indeed, nondiscrimination rules are applicable to groups generally, even apart from any benefits they get; much depends on whether the groups are seen as small and selective enough to be covered by a right to "intimate association," and on whether apply antidiscrimination law to the groups would interfere with the groups' expression of their ideas, and thus burden their right to "expressive associations." See Roberts v. U.S. Jaycees (1983); Boy Scouts of America v. Dale (2000). The university might thus be able to discipline students who (a) are involved in a fraternity's admissions decisions, and (b) can be shown to have denied membership to people based on race, or intentionally tried to communicate to potential members that they would deny them membership that way. I don't think that a discussion saying that discrimination ought to take place, or even that at some unspecified time it will take place, would suffice to constitute a violation of the antidiscrimination rules, though it might be used as evidence in a future case where discrimination against a particular applicant might be alleged.
But even if the group is found to have discriminated against black applicants, and some particular members were found to have participated in that decision, the penalty for that has to be based on the penalties that are actually meted out to people who violate this rule. If discrimination by a group generally leads to a fine against the group, or a reprimand of the participants, or even derecognition of the group, the university can't then expel students who engage in the same action but who also engage in constitutionally protected speech - that sort of disparate treatment shows that the school is really punishing people for their speech, not for their conduct.
This is a familiar principle from antidiscrimination law: if a black student is expelled based on conduct for which white students are generally just mildly reprimanded, the law recognizes that the expulsion was based on the student's race, not just the student's punishable conduct. The conduct in that situation is being used in large part as a pretext for race discrimination. Likewise, if SAE members are expelled based on conduct for which people who didn't engage in SAE's speech would generally just be mildly reprimanded, the expulsion would be based on the speech, not the members' punishable conduct, which would just be pretext for punishing students for the ideas they were expressing to each other.
5. Of course, this just applies to the university. It certainly makes sense that the national fraternity may suspend the student chapter, and that other fraternity or sorority organizations refuse to deal with the chapter (or even its students). Fraternities, at least in principle, aim to promote certain principles of morality and behavior, such as the national SAE's True Gentleman creed:
The True Gentleman is the man whose conduct proceeds from good will and an acute sense of propriety, and whose self-control is equal to all emergencies; who does not make the poor man conscious of his poverty, the obscure man of his obscurity, or any man of his inferiority or deformity; who is himself humbled if necessity compels him to humble another; who does not flatter wealth, cringe before power, or boast of his own possessions or achievements; who speaks with frankness but always with sincerity and sympathy; whose deed follows his word; who thinks of the rights and feelings of others, rather than his own; and who appears well in any company, a man with whom honor is sacred and virtue safe.
SAE may quite rightly insist that people who so sharply depart from such principles no longer use SAE's name. (I don't think a university may suspend a fraternity just based on its speech, but that question is likely rendered moot by national SAE's actions here.) Likewise, I imagine that the fraternity members' speech will more generally affect their social lives and their professional lives, as some people choose not to do business with them in the future. (In some states, even private employers are limited in their ability to discriminate against employees or job applicants based on their speech, but that's true only in some states and generally only as to employment; and, rightly or wrongly, such discrimination often happens without the applicant's even knowing that it's happening.) How long this sort of misbehavior should dog a person is an interesting ethical question, but in any event it's pretty clear that the offending students are going to pay a substantial social and likely economic price for their actions.
Under the First Amendment, though, the government - including the University of Oklahoma - generally cannot add to this price, whether the offensive speech is racist, religiously bigoted, pro-revolutionary, or expressive of any other viewpoint, however repugnant it might be.