Wouldn't that be a bit pointless, considering the Western Empire ended in 476 and Constantinople fell in 1453?
The Roman Empire became more of an idea than a country after late antiquity.
Technically, Franz II dissolved the entirety of the HRE rather than abdicating the throne, so the Pope would have to entirely reconstitute the Holy Roman Empire and there aren't exactly that many Catholic monarchs left. As I recall, Luxembourg, Liechtenstein, Belgium, Monaco and San Marino are still Catholic, as are of course Andorra and the Vatican City, but they wouldn't qualify anyway.
Why? Charlemagne didn't rule over a legally-defined area, he just conquered a bunch of places. Leo apparently gave him the title because he viewed the Eastern throne as unoccupied, so I don't think it would be necessary for a new emperor to rule over the territory claimed by the last. And even if it was, why couldn't someone be designated Emperor-in-exile (or be given a square foot of land from the Vatican to rule)?
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