I think the reason the staff breaks is because

1. 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. 

2. 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.