I think the reason the staff breaks is because
Sauron has infused a good amount of his power into the ring that the Witch King bears, and since Sauron is much much much more powerful than Gandalf, he could have given the Witch King enough power to be more powerful than a normal Miar.
The staff is only a part of Gandalf's power, and Gandalf is not allowed to use his full power while on Middle-earth, only a small fraction of it. If he could use his full power, he could have destroyed Smaug, instead of getting a bunch of Dwarves to do it. The Balrog that Gandalf killed was probably much weaker than a Sauron-aided/power-induced Witch King.
Therefore, for these reasons, mainly because the Witch King was heavily power-fuelled by Sauron, it is very likely that the Witch King can hurt Gandalf. But who becomes the victor in a one-on-one battle remains to be seen.
