(CNN) Thrusting himself into the combative 2016 presidential campaign, Pope Francis said Thursday that GOP front-runner Donald Trump "is not Christian" if he calls for the deportation of undocumented immigrants and pledges to build a wall between the United States and Mexico.

The Pope, who was traveling back to Rome from Mexico, where he urged the United States to address the "humanitarian crisis" on its southern border, did not tell American Catholics not to vote for Trump.

But Francis left little doubt where he stood on the polarizing issue of immigration reform.

"A person who thinks only about building walls, wherever they may be, and not building bridges, is not Christian. This is not the gospel," the Pope told journalists who asked his opinion on Trump's proposals to halt illegal immigration.

Trump immediately fired back, calling Francis' comments "disgraceful."

"No leader, especially a religious leader, should have the right to question another man's religion or faith," he said in statement. Trump added that the government in Mexico, where Francis spent the past five days, has "made many disparaging remarks about me to the Pope."

"If and when the Vatican is attacked by ISIS, which as everyone knows is ISIS's ultimate trophy, I can promise you that the Pope would have only wished and prayed that Donald Trump would have been president," Trump said.

By Thursday evening, the GOP candidate had softened his tone.

"I don't like fighting with the Pope," Trump said at a GOP town hall debate in South Carolina hosted by CNN. "I like his personality; I like what he represents."

He added that he thinks Francis' remarks were "much nicer" than the media reported and that the Pope had been misled by Mexican officials.

Trump also said that the Pope has an "awfully big wall" himself at the Vatican.

That may be true, Catholic priests said, but Vatican City also has an awfully big door.

The tussle between Trump and Francis -- two outsized personalities who seldom shy from speaking their minds -- seems to have been building for some time. Before the Pope traveled to Mexico, Trump cast the pontiff as a political naif who "doesn't understand the dangers" at the U.S.-Mexican border.

Trump social media director Dan Scavino suggested the pontiff's comments were hypocritical. "Amazing comments from the Pope- considering Vatican City is 100% surrounded by massive walls," he tweeted.

Amazing comments from the Pope- considering Vatican City is 100% surrounded by massive walls. pic.twitter.com/g3iVLDVGe5 — Dan Scavino (@DanScavino) February 18, 2016

During the wide-ranging press conference aboard the papal plane, Francis also seemed to suggest that contraception may be used to prevent the transmission of the Zika virus and praised Saint John Paul II's "holy friendship" with a Polish woman

But it was his comments on Trump that seem sure to dominate the political conversation, perhaps handing a gift to Trump's GOP opponents and opening Francis to criticism that his papacy is too partisan and his policies too liberal. Polls indicate that while Democrats adore the Pope , Republicans view him a little less favorably.

Asked whether American Catholics should vote for Trump, Francis demurred.

"As far as what you said about whether I would advise to vote or not to vote, I am not going to get involved in that. I say only that this man is not Christian if he has said things like that."

The Pope appeared somewhat unaware of Trump's exact stance on illegal immigration, though, saying that he would give him "the benefit of the doubt" until he had heard exactly what the billionaire businessman had said.

The Rev. Federico Lombardi, a Vatican spokesman, said Tuesday that the Pope knows "Trump expresses himself in an expressive way," but "is not always up to date on the latest statements."

Trump has pledged to build an $8 billion wall along the United States' southern border and says he will force Mexico to pay the tab. Trump has also said that, if elected president, he would eject some 11 million undocumented immigrants from the country.

"You have people coming in, and I'm not just saying Mexicans -- I'm talking about people that are from all over that are killers and rapists, and they're coming into this country," Trump told CNN's Jake Tapper last June.

JUST WATCHED What it would take to build Trump's border wall Replay More Videos ... MUST WATCH What it would take to build Trump's border wall 02:34

Before the Pope left for Mexico, Trump called Francis "a very political person" and suggested that the pontiff, who celebrated Mass Wednesday near the U.S.-Mexican border, was a pawn of the Mexican government.

The Pope made light of Trump's accusations.

"Thank God he said I was a politician because Aristotle defined the human person as 'animal politicus.' So at least I am a human person," he said. "As to whether I am a pawn, well, maybe, I don't know. I'll leave that up to your judgment and that of the people."

The White House weighs in

White House spokesman Josh Earnest, asked about Pope Francis' comments on Donald Trump, referred back to President Barack Obama's remarks last month at the National Prayer Breakfast.

He said Obama "talked about how his own personal Christian faith informed his view of the values and priorities that he has chosen to champion in the White House."

"A number of those values and priorities are not shared by Mr. Trump," Earnest said, before taking a jab at Trump's questioning of Obama's Christian faith.

"I will however extend to Mr. Trump the courtesy he has not extended to the President and not use this opportunity to call into question the kind of private personal conversations he is having with his God," Earnest said.

The Pope in Mexico

The Pope's comments on Trump came on his way home from an emotional trip to Mexico, where the first Latin American pontiff was greeted by boisterous crowds that often burst into songs or tears as he approached.

Celebrating Mass on Wednesday in Ciudad Juarez , a city just across the border from the United States, Francis delivered a stinging critique of leaders on both sides of the fence, calling the "forced migration" of thousands of Central Americans a "human tragedy" and "humanitarian crisis."

"Being faced with so many legal vacuums," the Pope said during his homily before a congregation of more than 200,000 people, "they get caught up in a web that ensnares and always destroys the poorest."

As he prepared to leave, Francis thanked Mexicans for opening their doors and their lives to him. "At times, I felt like weeping to see so much hope in a people who are suffering so much."