Girls don't only give you eye contact and approach invitations in public just because you look good. They can sense your confidence, dominance and social skills by your body language and by the way you interact with others. A socially skilled guy or girl can peg a person as cool or as insecure from a mile away.
I have a wing who looks like a male model. Six foot four, good body, perfect face, great hair. He seriously looks like an Abercrombie and Fitch model. But, he barely ever gets eye contact because girls can sense he's insecure so easily. This overrides his good looks.
I have another wing who's a top-form PUA, and he gets AI all the time. This is despite the fact that he doesn't look good at all. So, while looks are a factor in how much eye contact you get in public, it's mostly determined by your inner game and confidence.
-Dan