We need to international agreement to stop hunting wild animal outside of the country territory. However, if even one nation will disobey it, it will not work. To make it work, we need to _force_ all nations to obey common law, i.e. we need central government with forces. It doesn't look doable without one or two world wide wars. And the global exchaution of resources is the fastest way to perform these wars.
So, to win, we need to create mess first, then clean up.
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.
So, to win, we need to create mess first, then clean up.