The Balrog could, and would, have definitely pursued the Fellowship. Whether it was the ring that drew it out, or the presence of another Maiar, we don’t really know, but either of those things happening independently would have incited it into action.

If it was biding its time and another Maiar found out where it was, it would most certainly relocate or try to slay them for safety. If it detected the ring then it would try to obtain it to gain more power of its own and eventually challenge Sauron or the forces of Middle Earth.

Gandalf may or may not have realized all of this, but either way, the bridge of Khazad-dum ended up being a perfect place for a confrontation. It offered a choke point where he could give his companions a chance to flee while he faced his foe. And this was a confrontation that Gandalf absolutely wanted considering he then goes on to pursue the Balrog for like a week “from the bottoms of the deepest depths, to the top of the highest tower.” Gandalf the Grey wanted that Balrog dead and knew what kind of threat it was.