Are you friends with any other moms/families that have children around her age? Perhaps she would feel more comfortable participating if you invited another family over to swim or to went out to a movie? That wouldn't put all of the pressure on her; it would be a family outing.

Has your daughter ever been bullied? If she has had negative experiences with other girls being mean to her (which she may feel too embarrassed to tell you about) that could also be a reason why she doesn't want to invite other girls over.

What about sending your daughter to a day camp, summer program, or even to a short sleep away camp? You could get some broshures and let her pick the one that interests her most (within yout budget of course). Then she would be in a structured activity with other kids (new kids) and in an environment where she might be more likely to make new friends with a clean slate.