There's a huge variation among religiously affiliated programs. Some are gay-friendly (faculty and residents) and very committed to reproductive health care. Others are much less friendly and have many faculty/residents who will not prescribe e.g. contraceptives because of personal beliefs.
ALL Catholic programs have to abide by the Catholic Council of Bishops' guidelines which essentially forbid vasectomies, tubal ligation, IUD placement, etc. There is more flexibility in terms of Depo and other contraceptives, though you are not going to be giving out free condoms in clinic. Some will forbid you to participate in abortions (even if off-site) while you are a resident. Others do not have absolute restrictions.
If this is an important issue to you, then you need to figure out where you stand and where the program stands. You also need to figure out whether you are comfortable in a work place where people do or do not profess their religious beliefs frequently and publicly.
There are nominally non-religious programs that I found to be even more conservative and restricted than some of the Catholic programs.
Just do your homework and be sure that you are going to a program that supports what is most important to you.
To start, notice that the Via Christi program page "What We Are Looking For" includes links to the Christian Medical and Dental Association (which has a very strong evangelical mission) and the anti-abortion Choices Clinic in Wichita. Also read their statement on their web page:
"Issues of faith are very important to a majority of the faculty and residents, and the religious philosophy of our residency and hospital system allows and encourages us to care for "the whole person" (body, mind, and spirit), the underserved, and those experiencing crisis pregnancies."
If this is what you're looking for, go for it.