The Greatest Pope title is debatable, just like Greatest President or Greatest Emperor. They are based on personal opinions, and are NOT set in stone.

TWT, your theory on the Catholic Church being a continuation of the Roman Empire is indeed intriguing, I have never heard it before. However, I would say that the Papacy has fallen as well. While the Catholic Church is still around, it no longer has anywhere near the power it once did. I would consider the Great Reformation to signal the start of the decline of the Papacy, but I am not exactly an expert on this particular time period.

On to my opinion of the posed question:
I believe it was mostly internal issues that brought Rome to its knees, and the Byzantines after that. Corruption in the government is often the cause of the fall of empires. This rule goes from the Roman and Byzantine Empires, to the Third Reich.

Neither the size of an Empire, nor its military can save it from having irresponsible, or otherwise incapable men at the helm.