Since it is a question of history and does not touch a matter of divine revelation, you can probably interpret it in whatever way you think the data leads you. But I will say that that is not how I interpret it. By my reckoning, the Western Roman empire fell before St. Gregory became pope, and large portions of Western Europe, including Spain and (the precursor to) France, already had established Catholic Churches before Gregory’s reign. Moreover, the “great Roman Empire” was headquartered in the Eastern half of the empire, in the city of Constantinople, and had been since the time of Emperor Constantine – and that half of the empire did not fall. It continued to exist until 1453, long after the Western empire was re-established under Charlemagne and Otto the Great. Because of this, by my reading, the Roman empire did not completely fall until Napoleon was defeated. Thus Pope Gregory the Great was, in my opinion, not the last of the popes under the Roman Empire, but Pope Pius VII was. But, of course, there are other readings of history.
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