Was the territory that Charlemagne ruled the Roman Empire?
Charlemagne was not so farsighted as to believe in the efficacy of a united Europe; rather he fought and worked to build a large and powerful Empire of his own, commonly known as the Carolingian empire. He was attempting to complete the empire which had been initiated by his father, Pepin the Short and his grandfather, Charles Martel. It was Martel who lent his name to the Empire, following his defeat of the Muslims at the Battle of Tours, which prevented further Islamic intrusions into Western Europe.
Charlemagne's association with all things Roman originates from the fact that he was crowned "Emperor of the Romans" on Christmas Day, 800 A.D. by Pope Leo III. There is some argument by historians that this was the beginning of the Holy Roman Empire, but there is no agreement on this point. It is more apparent that Charlemagne received the title in exchange for his having extricated the Pope from a difficult situation.
Bottom line, Charlemagne did not consider himself (nor has history considered him) as Roman in anything other than in name only after his coronation by the Pope.
No, the territory was not the Roman Empire either in name or in fact. However, Charlemagne is credited with reviving the idea that all of Europe should be united under one ruler and he came closer to realizing this goal than any other medieval ruler.
The territory that Charlemagne ruled was very extensive indeed. It covered essentially all of Western Europe except for the Iberian Peninsula and Southern Italy. However, it paled in comparison with the Roman Empire at its height. The Roman Empire, of course, had ruled the Iberian Peninsula and Southern Britain. It had ruled all of Italy and its territory had extended through the Levant and across Northern Africa. It did not control as much of what is now Germany and Poland as Charlemagne did, but it controlled more territory overall.
Charlemagne's idea of a united Europe was based on the Roman Empire, but he did not rule as vast of an area as Rome did at its height.