I also think Catholic universities have done a better job of trying to continue their missions while finding a way to be inclusive to their increasingly diverse populations. In addition to the ones mentioned above you have DePaul (which has almost 15,000 undergrads), University of Dayton, and the University of San Diego, all of which appear to be thriving. Having said that, Catholic schools generally are in cities so that tends to add to the inclusive feel and community engagement. I saw this from the perspective of a non-Catholic (and basically non-religious) person whose daughter attends an all-girls Catholic high school.