No, developers do not restrict occupants based on religion, race, or national origin. That has been illegal in the US at least since the 1964 Civil Rights Act.
The Ave Maria people certainly know this. These liberal critics are attempting to discredit Ave Maria before it even starts by portraying it as discriminatory.
However, he can prevent the establishment of such institiutions as "gentlemen's bars." or other places of low repute. The problem is that as time wears on, the original owners of homes are replaced by persons of a different stripe.
There are various ways of doing this however, while having a theoretically open process. For example, in the upper class neighborhood of Chestnut Hill, here in Philadelphia, about 1/3 of the housing is owned by a trust created by the developer of the neighborhood. These houses are rented out by an agency he founded that personally vets all of the prospective applicants, most of whom, of course, are referred to the agency by existing residents, former residents, members of the neighborhood country club, etc. as being the "right sort of people" to join the neighborhood. Sure, the riff-raff might try to apply for a unit, but I've never seen someone of that sort actually get one.