Mr Trump said it was "disgraceful" for a religious leader to question a person's faith.
Who is Donald Trump?
D.O.B: June 14, 1946 (age 69)
From: Queens, New York City, U.S.
Job: Chairman and president of The Trump Organization, Chairman of Trump Plaza Associates, LLC, Chairman of Trump Atlantic City Associates
Political party: Republican
Worth: $4 billion
The pontiff was making the comments in relation to the Republican's action to address illegal immigration - by building a wall along the US-Mexico border.
His comments came hours after he concluded a visit to Mexico, where he prayed at the border for people who died trying to reach the US.
While speaking to reporters on the papal plane, he was asked what he thought of Mr Trump's campaign pledge to build a wall along the entire length of the border and expel millions of people in the US who had come there illegally.
"A person who thinks only about building walls, wherever they may be, and not building bridges, is not Christian," he said.
But Pope Francis said he would "give the benefit of the doubt" because he had not heard Mr Trump's border plans independently.
He added: "I say only that this man is not a Christian if he has said things like that."
Mr Trump, a Presbyterian and the front-runner for the Republican presidential nomination, responded within minutes: "For a religious leader to question a person's faith is disgraceful."
"I am proud to be a Christian, and as president I will not allow Christianity to be consistently attacked and weakened."
Donald Trump also raised the idea of the Islamic State extremist group attacking the Vatican, saying that if that happened, "the pope would have only wished and prayed that Donald Trump would have been president because this would not have happened".
Later the billionaire businessman said he was "totally respectful" of the pope but stood by his initial response.
Before Thursday's exchange, Donald Trump had been critical of Pope Francis's visit to Mexico.
He said last week that the pope's plans to pray at the border showed he was a political figure being exploited by the Mexican government.
However, Peter Smith from Catholic Voices told Premier the Pope was not just referring to Donald Trump.
Speaking on Breakfast he said that he was talking about "those who only build walls and never build bridges should be castigated, and I think all evangelical Christians would agree with that.
"Trump has not done anything to talk about the ways that migrants in the US could be settled, or how the problems of flows of migrants from Mexico to the US could be alleviated."
Listen to Premier's Rosie Wright and John Pantry speak to Peter Smith here: