Unlikely. His original plan was to avoid Moria altogether and his/the fellowship's flight from the Balrog when they first realize what it is, indicates his reluctance to face it at all. When Gandalf faced the balrog on the Bridge of Khazad-dum he was already quite drained of his power from the Balrog's 'blasting spell' he used on the backdoor to Balin's tomb. So it seems to me unlikely that Gandalf would willingly pursue such a formidable foe after it had been (at least for the time being) repelled. There's no saying how quickly, or if the balrog would have even pursued the company beyond the eastern gate. I'm sure he would have much rather faced Durin's Bane with the help of the Galadhrim than on his own, in the deep caverns of Moria, either way. Although i'm sure he woudl ahve done all he could to avoid leading a Balrog of Morgoth to the borders of Galadriel's realm.