After completing his doctorate, he served in parishes and as a lecturer. From 1987 to 1993, he was rector of Chesters College, Bearsden.
In 2004, he was appointed rector of the Pontifical Scots College. A year later, Pope Benedict XVI named him Bishop of Paisley. He was ordained bishop on Nov. 20, 2005, by Archbishop Mario Conti.
He took the motto "Da robur, fer auxilium" ("Thine aid supply, thy strength bestow"), from St. Thomas Aquinas' Eucharistic hymn "O Salutaris Hostia."
Benedict XVI named Tartaglia archbishop of Glasgow on July 24, 2012. He succeeded Conti, who led the archdiocese from 2002 to 2012.
Tartaglia was installed at St Andrew's Cathedral, Glasgow, on Sept. 8, 2012, the Feast of the Nativity of the Blessed Virgin Mary.
He spoke out last year about forced evictions of refugees and asylum seekers in Glasgow, describing them as "regrettable and harsh."
"I appeal to you not to make refugees and asylum seekers homeless, but to provide for them decent accommodation in accordance with their human dignity and human rights," he wrote to the U.K.'s Home Secretary.
Cardinal Vincent Nichols, archbishop of Westminster and president of the English and Welsh bishops' conference, said: "I have learned with great sadness of the sudden death of Archbishop Philip Tartaglia. All of Scotland will be saddened and shocked by his death, sentiments shared throughout England and Wales, too. He and his family are much in our prayers."
"I have often enjoyed the warm hospitality of Archbishop Tartaglia and admired his pastoral sense and sharp mind. His leadership will be greatly missed."
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Celtic F.C., a soccer team founded in Glasgow in 1887, paid tribute to the archbishop on its official Twitter account.
"We are saddened to hear of the death of Archbishop Philip Tartaglia who was a huge supporter of the club and regularly attended matches at Celtic Park," the club said.
"Everyone at Celtic offers their sincere condolences to Philip's family and Scotland's Catholic community at this sad time."