I can’t be the judge over each university and college, to say which ones are still Catholic and which are not. I consider that it is a matter of degrees. But it seems clear that the vast majority of Catholic universities and colleges have progressively moved away from their Catholic identity.
A university is not so much the buildings, as it is the people. No matter what the written documents say, which establish its structure and rules, the university is whatever its people are. And people have free will.
The leadership at many Catholic universities have decided to hire non-Catholic teachers and to admit non-Catholic students. That’s not such a bad decision. We live in a pluralistic world, and a Catholic school can reasonably choose to allow a pluralism in its faculty and student body. But when that is the case, the school will lose its Catholic identity, unless there are limits to this pluralism.
A Catholic university should not allow faculty to teach or promote ideas contrary to definitive Catholic teaching. You might suppose that this is already true in many institutions, but it is not. Even the theology faculty at Catholic universities (and also at seminaries) have strayed from magisterial teaching, apparently with full knowledge and deliberation. They openly oppose Catholic teaching, and stridently argue for contrary views. This has been going on for many decades now.
A mere rule on this subject is not sufficient. There should be a process for a student or faculty member to complain about teachers who oppose magisterial teaching, and for those teachers of falsehoods to be removed.
But then another problem arises. Some false teachers go so far as to claim that their grave doctrinal errors are merely the proper understanding of the teaching of Tradition, Scripture, or the Magisterium. And this problem occurs on the right as well as the left. It simply is not true that the conservative theological opinion must be the correct answer to any question.
Many Catholic professors of ethics have a poor understanding of magisterial teaching on the very subject they teach. Others understand well enough to know that their own teaching contradicts the teaching of the Magisterium.
I favor allowing non-Catholics to be students and faculty members at a Catholic school. But there should be limits. If most of the student body is non-Catholic and non-practicing Catholics, then all the grave sins found at purely secular schools will be found at Catholic schools. And you can remove the word “if” from the previous sentence. It is already the case. The vast majority of administrators, faculty, and students at most Catholic universities and colleges are not practicing Catholics. More disturbingly, some practice the Faith externally, while openly rejecting magisterial teaching on grave matters of faith and morals.
Sometimes the faithful need to draw apart from the world in order to strengthen their faith. So there should be some Catholic colleges and universities who only admit practicing Catholics as students, and only allow practicing Catholics to teach there. The administration should also be comprised solely of practicing Catholics. Of course, we are then left with the problem of defining who is a practicing Catholic. If you attend Mass regularly, but never go to Confession, I don’t think you are really a practicing Catholic. But you might disagree.
Please take a look at this list of my books and booklets, and see if any topic interests you.