[Eldar14]

Maybe just saying that he weilded the Flame of Anor added to his power. After weathertop, didn't Aragorn tell Frodo thet what he yelled at the the Nazgul ("Gilthoniel! O Elbereth!" or something like that) did more damage to them than anything else? So maybe it applies here, where just mentioning the Flame of Anor can both hurt and frighten the Balrog.

Sidenote: Could the orcs even have found the ring? I can't remember exactly, but I think that Sam told Frodo that he hadn't seen Galadriel's ring, but just a bright light. And anyway, if Gandalf had been defeated, the Balrog would have recognized the the ring without Gandalf's help.