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Primarily it is Protestant catholic bashers who believe this, though there is a tradition of this belief in the Greek Orthodox church. A fameous, scandalous telling of the tale (titled "Pope Joan") was translated to English by Lawrence Durrell, who, as an author, excelled in scandalous tales of incest, etc.
The tale persists without historical backup because anything that raises doubt about the validity of apostolic succession in the Roman Catholic church will delight someone who dislikes the RC church.
One of the reasons for believing it's probably not true is that the allegation of the event doesn't appear in the historical record until 4-500 years after it allegedly occurred. It's hard to believe that a 'pope' giving birth during a procession would have gone unremarked for hundreds of years, and then conveniently been noticed along with the development of Protestantism.
I'm not a Catholic myself, just someone who hates to see religious prejudice rear its ugly head in our world.