Let's begin with what the First Amendment actually says and how that is interpreted by the Supreme Court. After doing a quick Google search on First Amendment, I found this site at Cornell which I found helpful, especially the discussion at the Learn more link.
Where the Amendment itself begins with a restriction on Congress, the discussion says that has been extended to all of the Federal Government and to State Government as well. Fine. So issue one is whether a university such as the University of South Carolina is part of State Government. If it is not, it would seem the First Amendment doesn't apply.
It seems clear that private universities are not part of State Government. So if the incident that Zimmerman leads with had happened at NYU, where he works, then it seems clear there would be no First Amendment issue at all and the school could suspend the student as the University of South Carolina did.
If that is right, an interpretation that treated private universities differently from public universities for application of the First Amendment seems possible, but also would be awkward, at best. Under an interpretation that treated both types of universities uniformly, the conclusion must be that the First Amendment prohibitions are simply not applicable.
The second issue seems to be whether the Carolinian Creed is a law, one that might abridge a student's freedom of speech. As I am more familiar with the University of Illinois than the University of South Carolina, I will focus on the Student Code from my campus. Below is the operative sentence from Part B of the Preamble.
These values include the freedom to learn, free and open expression within limits that do not interfere with the rights of others, free and disinterested inquiry, intellectual honesty, sustained and independent search for truth, the exercise of critical judgment, respect for the dignity of others, and personal and institutional openness to constructive change.
At issue here are the highlighted phrases, which do constitute some limits on speech put into place for the good of the campus community. If the University is part of State Government and the Student Code is law, then there is a First Amendment violation. If the Student Code is not law, the First Amendment prohibitions don't apply. Here, each interpretation seems possible to me. The university cannot fine or imprison a student for violating the Student Code, but it can suspend the student or dismiss the student outright. One thing that matters a lot, in my view, but is not in the language explicitly is on the issue of intent when the rights or dignity of others are violated. Malicious intent requires a different treatment from a clueless trespass. The latter demands education rather than punishment and that is the way it is typically handled on campus. Most law doesn't accommodate the violator that way, which is perhaps one reason to think Student Codes are not law.
The last issue for me is place. In reading the Inside Higher Ed piece I immediately thought of my own classroom. Students do not have Freedom of Speech there. The norm for behavior is that a student can talk up if she first raises her hand and then is called on by me. If the student says something off point, I might very well interrupt. I want to note this rarely happens. Much more common is the issue that no student raises a hand when I would like some student response. Nonetheless, the point is that the instructor is armed with the usual tools to thwart disrespectful speech in the classroom.
A student speaking out on the Quad is an entirely different matter. That is a public place and the First Amendment is probably applicable there. But once you differentiate space where the First Amendment does or does not apply, there is then the issue of which sort of space is a Campus study room. Would the answer depend on whether the room was only accessible to people with campus ID cards? Our Library and our academic buildings have no such access restriction during normal hours, but our gym facilities do.
Much of the rest of Zimmerman's essay is a lament that infringement on the respect for others seems to matter on grounds of race but not on grounds of religion. In other words, the codes are enforced selectively. This is not a First Amendment matter, but it is a serious concern. However, given that the Salaita case is still pending, it would seem that if we think of these codes as applying to the entire Campus community rather than just to students then grounds of religion do in fact matter. If so, the real issue is what constitutes breach and what counts for acceptable criticism within civilized discourse. Drawing that line is undoubtedly difficult, which is why everyone seems to be struggling here. But that is not sufficient reason, in my view, for an argument that no line should exist whatsoever, which is what Zimmerman argues.