These geographical empires are not to be confused with the church hierarchies, which is not to say there isn't significant overlap and interplay between the two. Concerning the church, we must now distinguish between the churches resulting from the Great Schism of 1054 (also called the East-West Schism or sometimes even the Schism of the East), when the church in Byzantium split from the church in Rome to form the Eastern Orthodox Church. Henceforth, there would be the Roman Catholics in the west and the Eastern Orthodox Church in the east.
This Great Schism is not to be confused with the Great Schism of 1378 (also called the Schism of the West), which involved only the western church -- i.e., the Roman Catholic Church. When Pope Gregory XI died that year, there was a dispute between the French and Romans as to whether his successor would be French or Roman and whether he would serve in Avignon or Rome. The chaos even resulted in three competing popes for a while, but the schism only lasted 40 years, until a new conclave elected Cardinal Oddone Colonna, from a Roman family, as Pope Martin V in 1417.
Finally, we should not confuse the Roman Empire, in any of its manifestations, with the Holy Roman Empire, which was more Germanic than Roman. It was hardly an empire; it was more a loose confederation of states located in central Europe, not the Mediterranean basin, which the Roman Empire dominated. Its connection to Rome is that the pope bestowed the title of emperor on Charlemagne in 800, though Otto I, crowned emperor in 962, is considered to be the first formal Holy Roman emperor. The Holy Roman Empire continued, with ebbs and flows, until 1806, when Napoleon dissolved it. You might better remember the Germanic nature of the Holy Roman Empire if you recall that it was considered the First Reich. The Second Reich began after the Franco-Prussian War (1870-71) under William I and his chancellor, Otto von Bismarck.
Confused yet? Well, just be glad I don't delve into this paradox: Many historians consider the absolutist emperor Napoleon Bonaparte to be a great liberal reformer.