Pence, a former altar boy who had a strict Irish Catholic upbringing but later converted to evangelical Protestantism, appeared to be referring to the fact that the rest of his family had stayed in the Roman Catholic Church and that his departure from the faith had put some strain on family relations.
Pence's talks with Francis in the papal library lasted about an hour, twice as long as the meeting between the pope and President Donald Trump in the same frescoed room in May, 2017.
"I want to extend the warmest greetings on behalf of President Donald Trump, who so enjoyed his visit here," Pence told Francis before reporters and photographers were ushered out of the room.
After the private talks, Pence introduced to Francis his wife Karen and the rest of the delegation, including U.S. Ambassador to the Vatican Callista Gingrich and her husband Newt Gingrich, the former speaker of the U.S. House of Representatives.
During an exchange of gifts after the private talks, Francis gave Pence a small white box with a medal inside.
"I didn't want to ask," Pence said with a laugh as the pope handed him the box. "This is for mom."
As he was leaving the papal library, Pence, still clutching the small white box, said: "Thank you, Your Holiness, you have made me a hero ... God bless you."
Francis gave Pence some of his recent writings, including his World Day of Peace message for this year.
Pence flew to Italy from Israel, where he had attended events commemorating the 75th anniversary of the liberation of the Auschwitz death camp.He was due to meet Italian President Sergio Mattarella and Prime Minister Giuseppe Conte before leaving Italy.