Well, the courts have already had a lot to say about this issue with regard to fraternities, country clubs, etc. The basic legal principle is that in general it is OK to have a club that is exclusionary, *unless* that club carries with it some sort of elevated social standing. Fraternities, for example, were (are) part of the old-boys club network that continues later in life with country clubs, etc. That's why, whatever Augusta National might think, no country club or fraternity would stand a real chance in court if a woman really wanted to join.
So the basic idea is that a woman excluded from a fraternity is excluded access to power, whereas the same cannot be said of a man excluded from a sorority. The situation is asymmetric. It would be interesting to see how this plays out in court. In some cases, the courts have ruled that a club/organization cannot exclude on the basis of gender, race, religion, etc., but as I said, in some cases they have allowed exclusion because it did not really result in the denial of any kind of social status....