In this recently uploaded video by CNN channel, it shows that it is not clear as to whether we should use the possessive apostrophe or not, though, Sue Mendelsohn, Director of the Columbia Unv. Writing Center, took part in the video and said, "There is no correct answer, but the common usage would suggest that, similar to the race card, it would be the woman card"card."
However, Ngram shows that the only usage there is is "the woman's card"card." And shouldn't "the woman's card" be the only correct usage since we say "women's health"?