The rare back-and-forth between pontiff and presidential candidate was the latest astonishing development in a US presidential race that has been roiled by Trump's freewheeling rhetoric and controversial policy proposals, particularly on immigration.
The front-runner for the Republican nomination, Trump has said that if elected, he will build a wall along the Mexican border from Texas to California and expel 11 million people who are in the country illegally. The Pope's comments en route home from Mexico came hours after he prayed at the Mexican border for people who died trying to reach the United States.
Asked what he thought of Trump's pledge, Francis said: "A person who thinks only about building walls, wherever they may be, and not building bridges, is not Christian.
Not having heard about Trump's border plans independently, Francis said he'd "give the benefit of the doubt." But he added: "I say only that this man is not Christian if he has said things like that."
Trump, a Presbyterian, shot back within minutes.
"For a religious leader to question a person's faith is disgraceful," he said at a campaign stop in South Carolina, which holds a key primary on Saturday. "I am proud to be a Christian and as president I will not allow Christianity to be consistently attacked and weakened."
He also said the Mexican government has disparaged him to the Pope and separately invoked the Islamic State group, saying that if it attacks the Vatican, "I can promise you that the Pope would have only wished and prayed that Donald Trump would have been president because this would not have happened."
While Trump has made his plans to build a wall along the US-Mexico border a centerpiece of his campaign, other candidates have similar positions. Texas Sen. Ted Cruz and retired neurosurgeon Ben Carson are among those who have explicitly called for construction of a wall.
Other Republican candidates, including Ohio Governor John Kasich, have called for enhanced border security and fencing but stopped short of backing a wall or other structure along the entire length of the border.
The US government has built some 650 miles of wall along the 1,954-mile U.S.-Mexico boundary.
Marco Rubio, a Catholic seeking the GOP nomination, said in response to Francis' comments that Vatican City has a right to control its borders, and so does the United States.
Rubio said he has "tremendous respect and admiration" for the Pope, but added, "There's no nation on earth that's more compassion on immigration than we are."
Trump has been critical of the pope's visit to Mexico. He said last week that Francis' plans to pray at the border showed he is a political figure being exploited by the Mexican government.
Asked Thursday if he felt he was being used as a pawn of Mexico, Francis said he didn't know.
"I'll leave that up to your judgment and that of the people," he said.
He seemed pleased to hear that Trump had called him a "political" figure, noting that Aristotle had described the human being as a "political animal."