You are of-course assuming that a) anyone will care to wage a world war over whales, b) the 'right' side will win such a world war and c) A victory in such a war will be any better than defeat with irradiated lands and oceans.

But yes, in principle you are right - a common world government will need a world war. Such a government will also, by necessity, need to be a un-elected tyranny in order to exist as a common world government.