I specifically stated that the domain here is "the area over which Rome was the hegemon". Which is less than the whole world, obviously (but also more than Europe; North Africa, for example). We're talking about the rest of the world today because USA is truly a global hegemon, though.

And I don't know how you define "fall", but the standards of living and things like technological knowledge, culture etc basically cratered for several centuries; it sure does sound like a fall to me. "Failed state" is a modern term that doesn't quite have a direct analogy, but even so what was going on in most Roman provinces (and ultimately even Rome itself) could be reasonably described as such - stable governance would also take centuries to reassert itself.