It nowhere says any such thing.
Never does it say that there is a false Church, nor that the Church has anything to do with the state of mystery Babylon. Arabia and Constantinople are upon seven heights founded too.
We aren't told it is Rome. In fact, it is not the Roman empire, since in John's time of writing the Apocalypse, the Roman empire was the one present. He spoke of an empire that had been, was not any longer, while he wrote, and would be again. The Roman empire lacked many of the eatren countries that the former empires had all had. But islam had all those and more, and came after the Roman empire, including in it a Persian kingdom, a Mongolian one, an Arabic one, and a Turkish one - four horns that followed on that little horn that is not a man, but has the eyes of a man (muhammad, the author), is smaller than a man, being the size of a book, and called in Daniel the little qeren (quran).
The greatest Eastern Christian Cathedral, named Holy Wisdom (Hagia Sophia), was abominated and made desolate, in the city built upon seven hills (now called Istanbul), and four horns (minarettes) were placed all around it. The Mass was ended there, and since those days, no Christian has worshipped there. It was named Holy Wisdom, after the ONE WHO is GREATER than wise Solomon. Solomon at one point recieved a money sum to the sum value of 666. We are told that to calculate the mark of the beast takes "wisdom". The muslims (the beast) placed four medalions up in Hagia Sophia, on which was the blasphemous name of allah. The three letters in the earliest manuscripts of the Apocalypse of John were stylistically written in such a way that they closely resemble the crossed swords of islamic jihad banners, and a mirror image of the name allah,
Suleiman I (called the "magnificent") was the tenth and longest ruling sultan (ruling horn number ten) of Turkey. He frequented the Hagia Sophia in its abomination of desolation. He went to war against the Saints, overcoming many of them in war. The sultans established in the same city from which the Congregation of Hagia Sophia (which has a name like a Woman) were exiled into the wilderness, the great harem, the mother of harlots, to which women - even kidnapped nuns - from every ethnicity in the old world were forcibly brought to be ravished. The heathenized city, Istanbul, partook in the genocide of whole races of the Saints, getting drunk on their blood. It was only from the area of the later resulting Turkey that John's Apocalypse mentions seven cities, saying that satan's throne is in one of them. The sultan had Jerusalem a while, and even the ancient city of Babylon.
Like Nimrod, the founder of Babel, the muslims recognized no national lots from GOD, and imposed themselves on all else, recognizing no established authority. Out of Nimrod's kin came many of the first Arabic tribes. Like the Babylonians, muslims rejected the TRINITY, and worshipped satan (who is one selfish person, not wanting to share glory with another in trying to displace the TRINITY). he likes to be called "the god", not the devil.
The quran contains the blasphemies that Daniel heard the little qeren speaking. Only muslims ever took Christian nations. Barbarians like the Goths invaded, but were assimilated. Islam had the world in its hand. It had Sapin, Sicily, parts of France and Italy, all the countries that were once Asia Minor, Greece, all of North Africa, all of the near East and Middle East, and nearly all the Far East, the Balkans, and all that became the Soviet Union, and the former Persian empire, which Rome had never even dented in all its glory days.
Also, I must say that the Vatican Hill was not part of the city formed on seven other hills, which we call Rome, and that Constantinople, not Rome, was the merchant's gem of the world. The sultan could ask for cold sherbet from another land, and it would come to him cold. The Turks controlled the the trade routes, and even the mediterranean sea for a while.
Anyway, the Vatican has once again split off from Rome, and will never again be part of Italy whose capital is still Rome.