One of those humorous questions I'd love a serious answer to.
For one, Rome didn't just keel over, it was a long and drawn out process over centuries, and even after the accepted date 476, there were still splinters calling itself "Roman Empire"...
And I truly hope that history will look back on the USA in the same way, and see how the decline didn't start (but certainly accelarated) in 2016.
It wouldn't at all be surprising if part of the US actually survives and prospers as an independent nation whilst other parts fall down to an Economic level that matches the wealth producing capability of their economic frameworks and their workers (I reckon a post-Oil independent Texas would be at basically the same level as Argentina).
Better even: an outcome such as Britain after the Empire - a long drawn fizzle from primacy into mediocrity with delusions of grandeur - is realistic and possibly the best possible one.
Even post Imperial Britain had periods were most had a pretty decent life, such as the one that followed WWII and the rebuilding of the country, though the societal structures that underpinned that have been progressively destroyed since Thatcher and the results are pretty visible by now.
American merely stopping being top dog ain't too bad, but some of the other possible outcomes can be pretty nasty and that's just the ones were one or more Democratic nations are what's left of it. Descent into Authoritarianism would be the really ugly shit, not just for America but also the rest of the World on account of all the nukes.
The Holy Roman Empire emerged centuries after the fall of the Western Roman Empire and had little direct institutional or cultural continuity with classical Rome. The "Roman" aspect was more of an ideological concept and an attempt to invoke the prestige of the Roman legacy and the idea of a unified Christian empire in the West. So despite the name it was fundamentally a different entity from the ancient Roman Empire.
Imo, the USA to the British Empire is more like the Byzanthine Empire to the Roman empire. They bloom for some time and then turn into the the Sick Man of Europe...
Well acquisition of power through military means was a Roman tradition since Sullas march on Rome in 88 BCE though, so technically they have just as much of a claim as Charlemagne, the HRE, and the Tzars. A better one even since they actually conquered "new Rome" and its people and held it. What matters though is that they did claim the title of Caesar just as the other self-proclaimed successors of Rome did.