Is there an English word for non-vegetable fruits?

It's actually the same in English. Unless you're talking to a botanist, it's a non-issue.

Fruit (MWD)

a dish, quantity, or diet of fruits live on fruit

See also

Is a tomato a fruit or a vegetable?

As far as cooking is concerned, some things which are strictly fruits, such as tomatoes or bean pods, may be called 'vegetables' because they are used in savoury rather than sweet cooking.

Or in the words of Miles Kington

Knowledge is knowing that a tomato is a fruit, wisdom is not putting it in a fruit salad.

I think one can use the term culinary fruit to distinguish it from the botanical fruit term.

See, for example, this question on cooking.se (thanks 0xFEE1DEAD), this question on biology.se (thanks @Palitschke) or this diagram which explains the difference between culinary and botanical terms:

(Taken from the Wikipedia Fruit article).