Robert Joseph Dwyer was born in Salt Lake City in 1908. An only child, Dwyer attended Wasatch School before entering Judge Memorial Grammar and High School. He left Judge Memorial, perhaps under the influence of his uncle, a Marist priest, to enter the Marist Novitiate at Staten Island, New York in 1926, but the following year transferred to St. Patrick Seminary at Menlo Park, California.

At age twenty-four, on 11 June 11, Father Dwyer became the first Utah-born priest ordained specifically for the Diocese of Salt Lake City. Extensive reading during his earlier years had nurtured his gift for writing. As editor of The Intermountain Catholic from 1934 to 1938, he launched an ancillary career of forty-one years as a renowned journalist. He was assigned to graduate studies in history in 1938 and entered Catholic University of American at Washington, DC, earning his Doctorate of Philosophy in 1941. His dissertation, The Gentile Comes to Utah: A Study in Religious and Social Conflict 1862-1890, was published in 1971.

Returning to the diocese, Father Dwyer served in almost every ecclesiastical capacity there. He resumed editorship of the diocesan newspaper from 1941 to 1946, though several controversial articles prompted his removal from the position again in 1950. Named rector of the Cathedral of the Madeleine in 1948, he was elevated to the rank of Monsignor in 1950 and appointed Bishop of Reno in 1952.

During a tenure of fourteen years in Reno, he constructed Manogue High School, remodelled St. Thomas Aquinas Cathedral and participated in the newly formed Nevada Council of Churches. But on 5 June 1961 Bishop Dwyer suffered a heart attack, although he recovered sufficiently by the time of Vatican Council II to attend four sessions of the Council in Rome. His appointment in December 1966 to the metropolitan See of Portland in Oregon came during a period especially challenging for the bishop of conservative bent.

With pen still always in hand, he served simultaneously as chairman of the National Catholic Register and the Catholic Twin Circle from 1967 until 1974, entering into post-Vatican discourse on doctrine and policy. Waning in physical strength, he resigned the office of Archbishop 22 January 1974 and withdrew to Piedmont, California. He died of cancer on 24 March 24.