Pope Primaries?

Unlike Western democracies, the government of the Catholic Church is more of a monarchy.
The Church is a hierarchy, in that the supreme authority is vested in one man, the pope. However, bishops control regional areas, known as dioceses, while priests run their individual parishes. Individual Catholics do not vote for their pope, bishop, or priest. The hierarchy operates more like a military chain of command than an elected democracy. In other words, no, there is no primary and no popular vote.

Other religions and Christian churches allow for the laity to participate in positions of authority from a little to a lot, but Catholicism has pretty much remained monarchial since the appointment of St. Peter. Laypersons are encouraged to participate in other ways,

You may have heard the phrase, "He who enters the conclave a pope, leaves a cardinal." The meaning? When a pope becomes sick or elderly (or dies), the College of Cardinals meet in a conclave to elect a new pope. During that time, rumors are rampant over who will be elected to take the Chair of St. Peter. Often, the media will name certain cardinals as the most likely candidates, called papabile (meaning pope-able in Italian). But usually, the papabile cardinals are almost never elected. So if a man enters the conclave as a favorite (or worse, acts as though he wants the job), then he'll leave as a cardinal because his fellow cardinals will choose someone more humble.