Quinn, I agree that a no-football Big East would not be interested in GW, but I don't that it is because "Georgetown would never allow it." I'm not aware of any provision in the Big East by-laws that allows one team to prevent the admission of a prospective new member. I agree that Georgetown would certainly oppose it & that's one vote against the. Georgetown would also lobby & would create alliances with other members based on mutual interests to form a voting block that would defeat a GW candidacy. As you suggest, Villanova would be a prime partner for such an alliance. Bottom line is that Georgetown would need some help to defeat such a GW candidacy.
I realize this is two years old, but basically, the Big East basketball five (Villanova, Seton Hall, St. John's, Georgetown and Providence) have historically voted as a group in virtually all matters.
All five are against GW joining the Big East. Because if Providence opposes GW, Georgetown in turn opposes UMass, Boston U or Rhode Island.
All five are categorically against teams that threaten their market standing.