It seems, that very moment was the time when little folk was in charge and great powers (warriors, kings, wizards, elves, wraiths, Sauron and even Valar) could just stand and watch. Of cause it was in the Music but I wouldn't describe it as Eru's intervention.
Gollum dancing on the edge shows us that his soul was not completely subjugated by The Ring as he still was able and free to do such a foolish thing! Imagine Sauron sitting there and saying: "What the hell you are doing! Stop it!!!".
But what made Gollum to fall? His fate? His fate was bound to The Ring. And I tend to think (Copiright
) that it was The Ring that made him to fall. Sauron put his malice into The Ring; he made it capable of bewitching and destroying whoever bore it. What he didn't expect, as it seems to me, is that The Ring's final "intention" was not only to return to its Master but to destroy him in the end - because self-destruction is inherent to evil as the example of Melkor shows. Dancing Gollum provided an excellent shortcut to destruction of Sauron and three others who used to bare The Ring, so The One didn't miss the chance. But it was neither a mere chance nor Eru-From-Machine.