I've always been under the impression that they could
Bah, the humanity's chances of destroyign all life on Earth are practically nil. Even many of the most resilient surface dwelling animals would survive a nuclear war, and when you go into oceans and all the bacteria and other stuff... you just can't kill everything.
I even doubt it would be possible to destroy human as a species entirely in a nuclear war. Oh, it would be possible to do significant short term damage to ecosystems and destroy multiple species that aren't able to adjust on changes on short notice - and species that take long to reach sexual maturity would suffer from radiation more than, those that take less time - but still, there's no snowball's chance in hell the nukes could manage - directly or indirectly - convert all biomass in Earth into deadmass.
It would suck ass majorly, but it would not be the end for life on Earth. Culture as we know it, yeah... many species, yeah, but life would go on.
Even if most mammal species - including humans - died out, we still have rats, omnivoric, venerable species that can adjust to pretty much everything. In about 50-70 million years there would probably be a new species capable of building nukes, evolved from rats (compare - dinosaurs died out about 65 million years ago and the evolution of land mammals began on it's earnest).
