Professional fraternities are often open membership- open to anyone who meets the requirements (such as a certain GPA, major, etc.). In terms of something like Phi Alpha Delta, all one has to do is fill out the application and turn in a check.
I think one thing to keep in mind is that all of these organizations are private organizations... they can discriminate as they please in most cases. For the same reasons that Boy Scouts can discriminate against homosexual members. If the organization gets funding from a goverment group, there would very well be a stronger cause of action.
If there was a cause of action, the burden of proof would NEVER fall with the GLO- that goes against the entire US justice system. Because race is a "protected class" it may be looked at using a different standard than if it was some other factor.
I'm not going to get into all the procedure involved, i'm sure i'd bore you all.
But those are the quick and dirty answers.