What criteria?
First of all — The Vatican is not on one of the seven hills spoken of in Revelation. It is across the Tiber.
Well, let's see what the vatican has to say about that.
"The ancient city of Rome was originally built on the topographical layout of seven separate hills. These seven hills were joined together by the ancient walls of Rome- the Servian walls. The Seven Hills of Rome lie in the east to the Tiber River. Later many more areas were annexed to the city. The city walls were extended to include them also and the city is now much bigger than its original territory. The Vatican and Vatican hill are included in the Areas that were added to the city much later in history." [http://vatican.com/articles/rome/the_seven_hills_of_rome-a4131]
The Catholics using that lame "across the Tiber" doesn't work. "The Vatican and Vatican hill are included".
First of all — The Vatican is not on one of the seven hills spoken of in Revelation. It is across the Tiber.
The significance of Rome lies primarily in the fact that it is the city of the pope. source: http://www.catholic.org/encyclopedia/view.php?id=10148
"Here is the mind which has wisdom. The seven heads are seven mountains on which the woman sits...Rev 17:9
ummmmmm.....
On the well-watered plain?