So, here's what I'm thinking, and not for the first time:
If Uncle Jesse is so super-Greek, and Danny Tanner has dark hair and brown eyes, why are all the Tanner girls so fair and blonde? Their mom (also, presumably super-Greek) is supposed to have been blonde with blue eyes?
I'm pretty sure I know the answer to this one, and it's not that some Greeks are fair and blonde. I know this is true, but the Katsopolis family is not from that part of Greece. (Have you seen Jesse's parents? Not blonde.) And it's not that blue eyes and blonde hair are recessive traits, which of course they are; even if somehow both Danny and his wife passed on the recessive gene for, say, blondeness, the chances that all of their daughters would be fair and blonde with blue eyes is very low.
No, in this case, the answer lies in the great sitcom tradition of "Not Giving a Shit." Some sitcoms are more concerned with this internal continuity than others, and I don't think I'm shocking anyone by observing that Full House and its writers/producers didn't mind making the impossible possible from season to season or episode to episode. For example: Joey Gladstone, comedian, was never funny. A hip, cool rocker like Uncle Jesse should probably not be obsessed with corny oldies music. And the Beach Boys don't come to your house and sing songs with you, no matter how cute a family you are.
My theory is that no one ever planned to explore Jesse's cultural heritage. As you may know, in the first season he was "Jesse Cochran." It was only in the second season that it was changed to Katsopolis, seemingly at actor/star John Stamos' request. This doesn't explain why they hired Stamos, who was always Greek with very dark hair and skin, to be the uncle of three blonde girls…but some things don't need to be explained. Some things can be chalked up to a sparkly eye and a winning smile. Maybe that's the real lesson here.