"not eating fish" is not the same as "Eats no fish."
In German, it is the same.
"Seine Freundin isst keinen Fisch." is also correct if we suppose that his girl is actually not his daughter but his girlfriend. Or is it wrong in English too?
Shouldn't this be a negative sentence like "Sein Madchen isst Fisch nicht".
What is the word girl doing here? In English the Genetive form is often used in connection with either girl friends or daughters, but not girls.