I don't think he would be reborn again, over and over. I think, if the Witch King had killed him that would have been it for Gandalf. The fact he was sent back first, was a rare occurrence indeed it wasn't even something that was done by the Gods ie the Valar.
Tolkien himself states that it was by a "higher authority" in letter 156. Tolkien goes out of his way to stress that the "Gods" were only concerned with the embodied world and its time, but Gandalf 'passed out of thought and time'
The authority higher than the Valar is only Eru, and Gandalf was only sent back due to the weight of his self-sacrifice battling the Balrog.
That being said, I do not think the Witch King could have beaten him. Whilst the two were close to even, Gandalf had, on many times acted as 'angel' in an emergency. To quote 156 again
"he does reveal a sudden power: he twice rescues Faramir. He alone is left to forbid the entrance of the Lord of the Nazgul to Minas Tirith, when the City has been overthrown and its gates destroyed-and yet so powerful is the whole train of human resistance, that he himself has kindled and organized, that in fact no battle between the two occurs: it passes to other mortal hands"
and on the nature of his sacrifice being accepted
"So Gandalf sacrificed himself, was accepted, and enhanced, and returned. 'Yes, that was the name. I was Gandalf.' Of course he remains similar in personality and idiosyncrasy, but both his wisdom and power are much greater"
The "much greater" wisdom and power would suggest to me that, if need be Gandalf was a match for the Witch King, even when he was enhanced. Whilst Gandalf, at one point does say along the lines of "Black is still mightier" I believe he is referring to Sauron himself. Rather than the Witch King. Gandalf is not immune to worry or anxiety so he would doubt his ability to strive against Sauron personally.
__________________ Yet the making of things is in my heart from my own making by thee; and the child of little understanding that makes a play of the deeds of his father may do so without without thought of mockery, but because he is the son of his father.
The Witch King vs Gandalf would be the ultimate battle in the Lord of the Rings but Tolkien never actually made it possible for one to determine if one could defeat the other. Gandalf the Grey fended off the nine at Weathertop but we don't know what happened there and Tolkien did make note that the nine at that point were more in a kind of scouting mode then battle.
To determine a true superior of these two is something only JRR Tolkien could have done and since he has passed away the answers shall never be known.
Gandalf has the capability to strive against Sauron, and deflects his gaze from the ring at a critical juncture shortly after his return to Middle Earth. The exact quote is in my thread, but essentially he gets into a test of wills with Sauron, which tires him out, but which he wins.
The details aren't given, but I suspect it is similar to when, in the Silmarillion, the She- Elf gets into what amounts to a song/magic battle. She sings a 'song of life and growth' against one of Morgoth's servant's 'song of dominion and death.' The details are hazy, so if anyone has a quote or reference please send/post it for me to see.
Deflects his gaze, strive against him by power of minds yes I agree. I meant strive in a more physical sense. In combat, power to power on a battlefield kinda thing.
__________________ Yet the making of things is in my heart from my own making by thee; and the child of little understanding that makes a play of the deeds of his father may do so without without thought of mockery, but because he is the son of his father.