Mr. Pence’s life has been organized around his faith, ideology and political ambition. “I’m a Christian, a conservative and a Republican — in that order,” he is fond of saying with his trademark look-you-in-the-eye earnestness. He handed out essays by Russell Kirk, the influential conservative writer, to his staff at the start of his governorship. He can recite part of a speech President Ronald Reagan delivered in Indianapolis, and he turned deeper toward Christianity after losing his second consecutive congressional race at age 31.

Mr. Trump, who has switched his party registration several times, has pointedly noted to those on his right politically that “this is called the Republican Party, not the conservative party,” unabashedly acknowledged that he gets his policy insight from watching television news programs, and has been unable to name a favorite, or any, passage from Scripture.

Mr. Trump has enjoyed political fortune by belittling his opponents, handing them demeaning nicknames, and even mocking their spouses.

Mr. Pence, on the other hand, so believes in the virtues of civility that he wrote an article, “Confessions of a Negative Campaigner,” in which he swore off the politics of attack after losing a congressional race. “Negative personal attacks have no place in public life and serve to erode public confidence in our basic institutions of government,” he wrote.