Appeal to the unconverted: The pope has reached out through words and deeds to Muslims, homosexuals, atheists, the disabled and women—groups of people who traditionally feel ostracized or marginalized by the Church. Gingrich compares the pope's efforts to Jesus Christ, who was "radical in meeting people where they were—washing the feet of his disciples, eating with tax collectors, letting prostitutes wash his feet with expensive oils, engaging with the adulterers at the well."

The GOP must be as inclusive, said Gingrich, who converted to Roman Catholicism in 2009. "Nobody is off limits." Writing off 47 percent of Americans who typically vote Democratic, many of them minorities "in its spirit is anathema to a good political leader," Gingrich said, a slap at 2012 GOP nominee Mitt Romney.

"It starts with meeting people where they are," Reed said of the GOP. "If somebody is hungry, feed them. If somebody is naked, clothe them."

Make the party more approachable: Pope Francis famously checked himself out of his hotel after his election. He owns a 1984 Renault given to him by an old priest. He stiffs security and mingles in crowds. He cold-calls ordinary people, such as the Italian woman who feared she would not be able to baptize her baby because it was born out of wedlock. "Francis is a happy warrior," wrote Feehery, "with a vision that focuses on the bigger picture, and he is determined to open the Catholic tent to embrace the world and make believers out of nonbelievers."

Republican consultant Scott Reed (no relation to Ralph) said the pope is a populist, extraordinarily attuned to a skeptical public. "I am not a Catholic but I'm watching this pope intensely and am immensely impressed with how he understands that people are turned off by institutions, including government—including the parties." said Reed, who ran Bob Dole's 1996 presidential campaign. "The pope stands for something. He doesn't just talk about it. He lives it and breathes it."

While Pope Francis dismissed the "bishop of bling," Franz-Peter Tebartz-van Elst, over extravagant spending, "Republicans are seen as the defenders of the rich and powerful instead of the poor and vulnerable," wrote Marc Thiessen, former speechwriter for President George W. Bush, earlier this year in The Washington Post.

Lead with humility. The pope said, "If one has the answers to all the questions that is the proof that God is not with him. It means that he is a false prophet using religion for himself. The great leaders of the people of God, like Moses, have always left room for doubt. You must leave room for the Lord, not for our certainties; we must be humble.''

Feehery said there is not enough humility in his party–or in Washington, for that matter. "In a world of egomaniacs like Rush Limbaugh, Glenn Beck, Mark Levin and Sen. Ted Cruz (R-Texas)—and Barack Obama—wouldn't it be wonderful to have a leader who left a little room for doubt, a little room for the Lord and maybe a little room to find honorable compromise to help make the government work?" he wrote for The Hill.