There was a fairly recent study on the impact of nuclear weapons on the ozone layer. It turns out it only takes a remarkably small number of nuclear detonations to do serious damage to the ozone layer.

So strictly from an explosive force impact nuclear weapons would not be able to destroy everything on the planet, but fire a few here and there across the world and you wouldn't need to. The UV radiation from the sun would effectively wind up sterilizing the surface of the planet.