The following is a quotation from an article from the aish.com Chabad Lubavitch website which explains this Judaic belief. I qualify this quotation by clarifying that the Bible does not identify Edom with Rome, much less Christendom or Roman Catholicism. This is, however, what the rabbis teach, and consequently, this is the foundation for the Noahide component of certain sects of Protestantism which associates Edom and Babylon with Roman Catholicism:
Esau, who embodies the power of might and sword, will, through his descendants, give rise to the Roman Empire or "Edom" as the Bible calls it. The power or Rome clearly lay its ability to conquer, dominate and build and Empire. Even after the decline and fall of the Roman Empire, the spirit and power of Rome will perpetuate itself through the rise of the West and the Empires of Europe. And, of course, it is the Romans (as in Roman Catholic Church) that converted the world to Christianity ...
As intense as the rivalry is between Isaac and Ishmael (the Jews and the Arabs) they are only half brothers. Jacob and Esau are twins with the same genetic material. This rivalry (Israel and Rome/The West) is understood to be the ultimate rivalry in history. This is nothing less than a cosmic struggle. These two -- Jacob and Esau -- started fighting in utero, and they're going to be fighting throughout history. The battle continues until today and does not end until the final showdown during the messianic era. It's not an even battle ever. Esau will always be stronger in the physical sense, but the Jewish people have inner strengths, resources and a destiny that will ultimately lead to their triumph ...
A few proof texts for this rabbinic teaching:
"...And a long time after him [Jesus of Nazareth], a religion that was attributed to him spread through the Children of Esau [i.e., Rome and its empire] ..." Moses Maimonides, Letter to Yemen
"They are called Edomites who move their fingers 'here and there'" (who make the sign of the cross). (Rabbi Bechai, Kad Hakkemach 20a)
"The Nazarenes are Romans, the sons of Edom." (Rabbi Abarbinel, Maschima Ieschua 36d)
"He who exacted vengeance from the former [oppressor] will exact vengeance from the latter. Just as in Egypt it was with blood, so with Edom it will be the same." (Rabbi Hama bar Hanina, Pesiqta de-Rav Kahana, sec. 7, 133)
I will remind readers that rabbis who promote and are guided by this vengeful, racial-supremacist psychodrama, were honored by the "Christian Conservative" President George W. Bush two days ago, April 15, 2008 for their contributions to the world in "education," and that the Vatican cannot stop praising and making accommodations to Orthodox rabbis whom Benedict will visit in a synagogue tomorrow, Friday, April 18, 2008. Meanwhile, the traditional Catholic liturgy is altered to suit the sensibilities of the rabbis who on this evening will remove the leaven from their homes in hopes that "Hashem," which is to say, the so-called "Jewish" people themselves, will annihilate us.