ROME — Pope Francis already has distinguished himself from his predecessor with a more down-to-earth style. Now he is both unnerving the Vatican and delighting the faithful by picking up the telephone and spontaneously calling people, earning the nickname “the Cold Call Pope.”

This month, he called to comfort a pregnant Italian woman whose married boyfriend had unsuccessfully pressured her to have an abortion. The woman, who is divorced and will be a single mother, wrote to the pope, fearing she had fallen afoul of the church. Not knowing the correct address, she marked the envelope “Holy Father Pope Francis, Vatican City, Rome.” The pope offered to personally baptize the baby when it is born next year, according to an account in La Stampa, a newspaper in Turin.

In August, Francis phoned a woman in Argentina who had been raped by a police officer. The pope told her that she was not alone and that she should have faith in the justice system, according to an Argentine television news report rebroadcast in Italy.

On Aug. 7, Michele Ferri of Pesaro, Italy, answered his phone and was startled to hear, “Hello, Michele, it’s Pope Francis.” Mr. Ferri said in a telephone interview he had thought it was a joke. “But then he spoke about the letter that I’d written, a letter I hadn’t told anyone about, not even my mother or my wife, and I knew that it was him,” he said.