Conan would cut Gandalf to pieces just like he did to every other wizard he ever met except Thoth-Amon. Thoth-Amon's foul sorceries are stronger than Gandalf's "tremble before my glowing staff!" petty conjurings, thus, I think Gandalf would be lunchmeat before the mighty Cimmerian's broadsword swung in earnest. :wink:
I have to disagree with you.
I would contend that if Gandalf can defeat a Balrog that wields a giant flaming sword and a whip, basically one of the demons created by Morgoth in the early days, then fighting a barbarian, however famous, would be somewhat anticlimactic.
Just because Gandalf chooses not to fight in most cases doesn't mean he can't whup a little @ss when required.
My wizard is better than your barbarian! nya nya nya nyaaaaah!
My real answer is if through some twisted cross-genre mixup Gandalf were pitted against Conan, he'd find some way to prevent actually fighting Conan in the first place. You know, maybe say something like "Hey, you're not my enemy, let's join forces and defeat <insert common foe here>." (something Conan does successfully quite often).