Erdogan is the first Turkish president to visit the Vatican in almost six decades.
After the French leader's remarks, Turkish Foreign Minister Mevlut Cavusoglu said that France had no right "to give us lessons" on a cross-border offensive, citing France's colonial history. No press conference with Italian media was held after the two official meetings.
The meeting today was the first in the Vatican between Francis and President Erdogan and takes place after a 29 December phone call between the two.
Erdogan and the pontiff "spoke about the situation in Turkey, the condition of the Catholic community, efforts to accommodate refugees and the challenges linked to this", the Vatican said in a statement after the audience.
The Vatican is in favor of a two-state solution to the conflict between Israelis and Palestinians and hopes that both sides reach a consensus on the status of Jerusalem - which houses sacred sites for Muslims, Jews and Christians.
Jerusalem is home to sites holy to the Muslim, Jewish and Christian religions, and Trump's recent announcement that he would move the US embassy to the city alarmed many allies, who say the decision could doom Middle East peace efforts.
The Turkish military suffered its highest number of casualties in a single day on February 3-4, when 10 soldiers were killed in a 24-hour period.
Pro-Kurdish demonstrators accused police of acting like Turkish security forces, saying they were being assaulted and corralled inside the Castel Sant'Angelo monument during the protest. Small-scale scuffles broke out between police and demonstrators nearby. At least one protestor was injured in the incident.
Turkey occupies a crucial strategic role as a bridge between the east and Europe but has faced criticism recently after launching a series airstrikes on Kurdish forces in Syria. The leader of right-wing, anti-immigrant Northern League, Matteo Salvini, said on Twitter it was "a shame" for the Italian government to receive the president of a "bloody regime".
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