Why does Sméagol eat live fish? Is it due to some sort of self-flagellation or guilt over killing Déagol, or is it merely that Sméagol lived an isolated life for 500 years, with fish as his only food source? It seems, in the movie, that Sméagol was actually revolted by regular food, such as the elven bread Frodo gave him and the rabbit stew Samwise prepared. So what's up with Sméagol and his haute cuisine? Does he have to eat raw fish, or is it just something he's become accustomed to?
In The Hobbit it says that Gollum also eats the occasional goblin when he can catch one out on its own. That's one of the things he uses the ring for.
He liked meat too. Goblin he thought good, when he could get it; but he took care they never found him out. He just throttled them from behind, if they ever came down alone anywhere near the edge of the water, while he was prowling about. They very seldom did, for they had a feeling that something unpleasant was lurking down there ...
And still sometimes he put [the ring] on, when he could not bear to be parted from it any longer, or when he was very, very, hungry and tired of fish. Then he would creep along dark passages looking for stray goblins ... Only a few hours ago he had worn it, and caught a small goblin-imp.
Gollum is revolted by Lembas (Elven bread) because most evil beings have a hatred for all things Elven. From the LOTR wikia,
"Like other products of the Elves, it was offensive to evil creatures; Gollum refused outright to eat of it."
As for the rabbit stew, I believe that after years of living in a cave and eating only raw fish, he balks at anything cooked, just like a wolf or other wild animal would if you gave it a stew made of potatoes and rabbit.
From The Two Towers:
Sam cooking rabbit stew.
Gollum: What's it doing? Stupid fat hobbit! You ruins it!
Sam: What's to ruin? There's hardly any meat on em. What we need's a few good tators.
Gollum: What's tators, precious? What's tators?
Sam: P-O-T-A-T-O-E, boil em, mash em, put em in a stew. Even you couldn't say no to that.
Gollum: Oh yes we could! Give it to us raw and wriggling! You keep nasty chips.
So you see, it's not the food, it's just the way it is served, in the case of the stew. He'd be happy to eat the rabbit, if it was raw. Gollum just prefers his food as a primitve cave person would.
I think it makes no sense at all from any logical point of view. Ask my dog if she minds eating my rabbit stew. Heck, I have to stop her from eating paper bags that smell like anything edible. But then, dogs are omnivores. Cats, on the other hand, are strictly carnivores, and will hardly eat day-old cat food.
Hobbits are, from all accounts, very much omnivorous. Assuming Gollum has gone feral in some way, would he then not eat potatoes to save his life? Of course he would. No, the answer must be that Tolkien portrayed Gollum as a creature corrupted by evil. Being in close proximity to the ring for all those years had changed his very core being, made him evil, in flesh and in spirit. What do evil monsters eat? Well, it's not potatoes. Blood and gore. Raw fish and baby rabbits.
The reason he would not eat lembas was that anything elven did not sit well with him; as you will recall, he was physically hurt when they tied an elven string to his leg. Presumably, the elven made objects carried some innate goodness that was harmful to Gollum's inner evil.
That's correct - he has lived in a cave with no fire for hundreds of years.
He set the pans down, and then suddenly saw what Sam was doing. He gave a thin hissing shriek, and seemed to be both frightened and angry. 'Ach! Sss – no!' he cried. 'No! Silly hobbits, foolish, yes foolish! They mustn't do it!'
'Mustn't do what?' asked Sam in surprise.
'Not make the nassty red tongues,' hissed Gollum. 'Fire, fire! It's dangerous, yes it is. It burns, it kills. And it will bring enemies, yes it will.'
He has grown accustomed - both psychologically and physiologically - to
sushi raw fish (or goblin... or other meat). Dharini's answer has all the juicy (no pun intended) quotes from The Two Towers for that occasion.
For a similar effect, consider any steak lover who likes his stakes rare, when faced with a "well done" one. NOT a pretty reaction from my experience.