I can agree that that would be a bad consequence. However, I didn't go to Notre Dame, Duke or Georgetown, but I assumed they offered a secular education, yet had a chapel, -pretty much like a Catholic Hospital. Liberty sounds like they are offering a religious education. I haven't been there, though, or known anyone that has.
This opens ever so many delightful cans o' worm. First, I'll go ahead and say that they are offering a "religious education," to the extent that such a thing exists, except ... for where they are not. And ultimately, they would deny (or at least most such schools would deny) that their religious affiliation in any way impedes the caliber of their schooling, or even affects the content of it. In some cases, I'm sure that's true, though in most, I wouldn't agree. The problem is that what would be necessary here would be to clearly prove some very fuzzy points. I think we'd have to stray all the way into the realms of metaphysics and epistemology in order to close the books on this one. (Which is to say, we never would). For example, what constitutes "religious education?" Harvard was, after all, a baptist school, originally. Almost all the venerable American Universities of note are (nominally) religiously affiliated as well. I think that U of M and U of Chicago are noteworthy exceptions, among others. Notre Dame arguably still has deep religious affiliations (and some of the attendant consequences, intended or otherwise), and Liberty isn't by any means occupying the far end of the spectrum. What percentage of courses offered, or curricula, need be religious in nature in order to classify a school as "offering a religious education?" And, clearly, we're not talking about whether the subject is religious, as any number of sociological or anthropological courses would fall into the category if so. And if we're discussing which didactic methods are acceptable, then musn't we pursue this inquiry across the spectrum to address politics of movement and systems of knowledge? Perhaps, if we're only discussing funding, not. Ultimately, I guess I don't care about the question where the matter of University is concerned (although it is very interesting politically, ideologically). Anything which restricts the latitude of its searches, whether dogma or funding strictures, is antithetical to the nature of the thing. But then again, as I've said before, the true form of the University is only incidentally and accidentally associated with the structure of the university as found in the world.
A lot of worms for sure. However, perhaps one reasonable test might be whether or not the institution trains its student's to practice a specific faith. Liberty's motto of "Training Champions for Christ since 1971." seems like their objective is a religious one. If Congress isn't to "make no law respecting an establishment of religion, or prohibiting the free exercise thereof." it seems that spending tax dollars on a organization in order to train champions for Christ would be a conflict. I think a religious institution can perform non-religious services, and that those services might be subsidized by the government, however, when the service is to advance the religion, the government should not help or hinder the organization. There is little doubt in my mind that if Liberty were an Islamic institute, the call for the separation of Church and State would be vociferous.
I see. Using an institution's mission statement would seem as good a method as any to designate the net purpose of an institution which is in reality manifold. I wonder what is in fact currently the measure? Also, it would be interesting to pursue another aspect of this further: in what way is the (non-state) university receiving federal money? Are you speaking solely of student financial aid money, or are there other matters? And isn't it technically the student which receives aid? I realize that this is a technicality and a simplification which might well be described differently. But upon such technicalities is policy built. In this case, the new big question would be this: what determines the set of institutions for which students may be aided financially? An Islamic institute should surely qualify if a Christian one does. However, even within Christian colleges, there is a divide between colleges and seminaries or "Bible colleges," i.e., schools which instruct only or mainly on matters of religion and theology. Here, as in the wider question of how to differentiate between schools of varying caliber and purpose, accreditation seems perhaps the best method. But I'm well out of my depth regarding what the standards in fact are.