Do you mean social media sites? People voluntarily give up that information to the public, so it could reasonably not be considered private.

The sex tape would be considered private.

Presumably he draws his line somewhere between those two extremes.

Has he done anything inconsistent with the position "The paparazzi should not write scandalous articles about famous people"? For example, has he invested in companies similar to Gawker? If not, I don't see any real conflict in his position.

I think most people consider what they post on social media sites to be for their friends, not the public. Of course, under an American moral definition of privacy, if you tell something to one person ever, it's as public as if you bought billboards across the country. And if you're tricked into releasing something to the public, say by a company which fails to properly advise you when you attempt to share things with your friends but actually publish them to the wider Internet, you have no recourse, legal or moral.