*sigh*
No, they don't.
Anyone can organize a Disney "group event" and get discounted tickets, etc. So-called "gay days" are organized and run by people outside the Disney organization. As long as the participants abide by the conduct policies Disney uses for all guests, Disney has no recourse to prevent them from doing this, even if they wanted to. (And I'll agree that they don't want to, as an organization, but that's beside the point.)
In theory, there could be a Free Republic day at Disney if someone wanted to organize it and file the paperwork for group discounts.