This seems like a valid and important question.
If the only inhibition, the only resistance to eating non-vegetarian food happens to be that it's bad/immoral/wrong because it involves cruelty to an animal and that it involves taking away the life of an animal, then it should logically follow that eating the [cooked] meat of animals which genuinely died naturally should be alright.
Now, the question that arises here is - how can this determination be made correctly, and in manner that can be consistently repeated - that a given animal died naturally and wasn't killed by man? This seems like a loophole that can be exploited rather easily. That being said, the mere existence of this loophole, this measurement difficulty doesn't take away the validity of the argument that cooking and eating naturally dead animals should be okay.