After completing his studies at the Roman Seminary he was ordained a Roman Catholic Priest by James Cardinal Hickey on December 17, 1971 in St. Peter's Basilica. From there he was assigned as Vicar Parochial of St. Francis of Assisi Church in Allentown, Pennsylvania where he also had duties as an associate professor at Allentown Central Catholic High School. In 1974, he was again sent to Rome to work on his doctorate in Canon Law from the Pontifical Lateran University, completing the program in 1977. In 1984, he was named pastor of Holy Trinity Church in Whitehall, PA.
On March 27, 1986, Bishop Welsh released Monsignor Callaghan to serve on the Vatican's Congregation for Institutes of Consecrated Life and Societies of Apostolic Life. Here he worked closely with Mother Theresa of Calcutta in canonical affairs concerning the Missionaries of Charity. In 1991 he was transferred to the Congregation for Bishops where he stayed until July 1995 when he became Vicar General of the Roman Catholic Archdiocese for the Military Services. He presently has been serving as rector of Saint Paul Seminary School of Divinity in Saint Paul, Minnesota since he was appointed to the position by Archbishop Harry Flynn on July 1, 2005.