I guess, from what I have read as I skimmed over this forum, that there is more going on with your daughter than just this, so maybe this isn't really applicable, but …
As a teenager I was incredibly unsocial, and very happy that way. I'd watch tv, or read a book or magazine, or listen to music. If I had to be outside I would watch bugs climbing on trees (not a joke, I really did that a lot, it was fun). I had no problem being around family, and maybe one friend (sometimes), but anytime I was forced into any kind of social situation I was utterly miserable.
Anyway, like I said, skimming over the forum and seeing your other posts, maybe there is more to how your daughter is than there was with me, but pushing her into being more social could very likely just make things even worse.