Suppose a very large asteroid or a cluster of large asteroids was headed towards Earth and we opted for the nuclear option to break a part the asteroids. Is a possible outcome that the asteroid debris, instead of burning up in our atmosphere (or impacting Earth), would instead: larger masses would form an asteroid belt around Earth; and space dust would clog up the atmosphere for a temporary "nuclear winter" type effect?

Is it possible that Earth could develop rings like Saturn from this event?