They said there was no game so they couldn't have eaten meat although there was that one episode where they somehow killed a pig for the Howells anniversary but I'm sure they did that just for the storyline since no other game was shown in other episodes.

There's no way the Skipper could've stayed so fat on just cocunuts, pineapples, berries, and bananas!! I've wonder if it is possible to only live on the sources they showed on the island and you could do it for a while but for years you would probably need beans or legumes of some sort for a better protein source. I know there was crab but a you'd still need more variety if you were stranded there for years.

A vegetarian diet is definitely ok since our closest primate relatives are over 95% vegetarian but a little more variety than just 3 or 4 fruits is needed!

There was already so many things on the island that you wouldn't find on a deserted island so they could have a storyline where a damaged boat or sub washed up on the island with tons of seeds for beans, legumes, lentils, carrots, spinach, rice, etc.

Then again there are frugivores and you do have all essential nutrients if you eat enough different fruits and YES fruits have protein so maybe it was possible on fruits alone if they had many different kinds.