[denethorthefirst]

Short Answer: No.
The world of Middle-Earth is entropic, i.e. everything becomes less grand and less powerful over time. This includes not only evil beings like the Umaiar but everyone in Ea, even the Valar become less powerful and diminish over time, not only because they, like Melkor, poured a lot of their Power into Arda during its creation, but because even they can't overpower this trajectory.

I fail to see how the Balrog, who at this point in the story (in the late Third Age) is already in a reduced and diminished (fully incarnated) state, should be able to accomplish something which even Manwe can't do. The only being in the story that "breaks" this rule is, curiously, Sauron who, after the making of the One Ring, got somehow more powerful. How exactly he achieved that peculiar feat ist never explained by Tolkien. The Extra-Power can't really come from the domination of the other Rings, because the Ring-scheme failed spectacularly (with the Elves and Dwarves). So where does all that Power come from? I speculate that the One Ring somehow allowed Sauron to tap into and control/usurp the "Morgoth Element", thats one explanation that would fit the nature of Ea.

In any case, that would not help the Balrog, because I don't think that he would be able to master the One Ring. Tolkien wrote that of all the characters in the story only Gandalf "MIGHT be EXPECTED" to master it, a remarkably vague and hypothetical statement by Tolkien (i personally don't think that Gandalf would've been able to achieve it). I guess that, if the Balrog somehow got possession of the One Ring, that he would become nothing more than a beefed up version of Gollum, becoming ensnared by the ring, retreating deeper and deeper into Moria, fleeing the outside world. After all, thats all the Balrog ever really wanted: to be left alone.