He graduated from Saint Charles College at Ellicott City, Maryland, in 1876, and then studied theology at Saint Mary"s Seminary in Baltimore.
He served as Bishop of Wilmington from 1897 to 1925. Returning to South Carolina, he was ordained to the priesthood on December 19, 1880. He first served as a curate at Saint Joseph"s Church and afterwards Saint Patrick"s, both in Charleston.
He was pastor of Saint Mary"s in Greenville (1882-1887) and pro-rector of the Cathedral of Saint John the Baptist and chancellor of the Diocese of Charleston (1887-1888).
From 1888 to 1897, he was assistant to the vicar general at Saint Patrick"s Church. On January 26, 1897, Monaghan was appointed the third Bishop of Wilmington, Delaware, by Pope Leo XIII. He received his episcopal consecration on the following May 9 from Cardinal James Gibbons, with Bishops Alfred Allen Paul Curtis and Henry Pinckney Northrop serving as co-consecrators.
During his tenure, he established seven parishes, seven missions, and eight schools. He also was instrumental in the establishment of the Oblate Fathers" Salesianum School for young gentlemen, Saint Francis Hospital, and a home for the aged.
After twenty-eight years as bishop, he resigned due to poor health on July 10, 1925.
He was named Titular Bishop of Lydda by Pope Pius XI on the same date. Monaghan later died at age 78.