Asking someone out through Facebook, text messages, instant messenger, phone, or anything that isn't in person is tacky. Breaking up with people that way is tacky, too
If this girl kinda likes this guy but isn't really sure, asking her out through any of the above means would totally be a turn off. Even if she really liked him, it'd probably be a total turn off.
Listen to Lunar. Asking her out over Facebook would be a disgrace to guys everywhere. And she WILL laugh about it for a looooooonnnnggg time with her friends.
Rin-chan