It appears to be a dying stereotype. When I looked for citations, the most recent I could find was from 1988. Anything more recent talked about how gender disparity in child care environments is generally seen as harmful.
On the other hand, to address the women's care centers .... thing, women STILL can't really walk into a doc's office or a hospital and get their lady parts checked out; they are required to go to a gynecologist. Whereas men, when they do have something weird going down with their bits, generally just go to a GP. It's getting better. I am no longer referred to my GYN for a simple yeast infection. However, there was (and still, to a point, is) indeed a need for separate women's care centers. It would be really nice if I could find a GP that was taking patients and could go in for my annual pap smear, for instance. Instead, I have to sit on a GYN's waitlist... and god help the women who prefer a female GYN.