"I do not eat fish."
Translation:Я не ем рыбу.
In Polish, you're correct. Objects in accusative case use the genitive case when the verb is negated. In Russian, however, an object in the accusative case stays in the accusative case when the verb is negated. Nevertheless, when the noun's presence is negated, hence its absence, then the noun declines to the genitive case. For example,
Рыба на столе. "The fish is on the table."
Нет рыбы на столе. "There is no fish on the table."
Суп с рыбой, "soup with fish" [instrumental case]
Суп без рыбы, "soup without fish"
Almost correct =) "I never eat any fish" would be in Russian "я никогда не ем рыбу", or if the sentence to translate more literally - "я никогда не ем никакую рыбу". "Я не ем рыбу" would be "I don't eat fish".
You may read my explanation of the using of this word which I wrote in my comments:
No, it is not correct for all russian nouns. Some Russian nouns are mass and other are not:
"I don't eat apples" would be "я не ем яблоки"