He studied at Saint Joseph Seminary College and Notre Dame Seminary.
He was ordained to the priesthood on June 12, 1932, at age 22. He then served as a curate at Saint Leo Church in New Orleans (1932–1946) and archdiocesan director of the Confraternity of Christian Doctrine (1937–1946). He was chaplain of the Newman Centers at Tulane University (1941–1946) and at Louisiana State University (1946–1959).
He was named a Papal Chamberlain in 1947 and a Domestic Prelate in 1949.
From 1954 to 1955, he was national chaplain of the Newman Club Federation. On March 13, 1959, he was appointed Auxiliary Bishop of Lafayette in Louisiana and Titular Bishop of Sergentza by Pope John XXIII. Tracy receivied his episcopal consecration on the following May 19 from Archbishop Egidio Vagnozzi, with Bishops Maurice Schexnayder and Louis Caillouet serving as co-consecrators.
He was later named the first Bishop of Baton Rouge on August 10, 1961, being installed as such on the following November 8. From 1962 to 1965, Bishop Tracy attended the Second Vatican Council.
On October 24, 1963, he addressed the Council in the name of his fellow American bishops on the subject of racial equality.
A year later, in 1966, he published his memoir of the Council, entitled American Bishop at the Vatican Council. He established a consultative process as an integral part of the diocesan administration, and encouraged the greater participation of the laity in governing the Church. Tracy also oversaw the construction of the Catholic Life Center and the renovation of Saint Joseph Cathedral.
In 1967, he became the first American Catholic bishop to publish a financial statement for his diocese.
In 1972, he established a committee for the regulation of allowing remarried Catholics to receive the sacraments, saying, "The Church has a pastoral responsibility of healing and forgiveness". He resigned as Baton Rouge"s bishop on March 21, 1974, after twelve years of service.
Tracy later died at age 70.