Barry was born in Oylegate, County Wexford, to Sylvester and Mary (Donohue) Barry, and while an ecclesiastical student in Ireland attached himself to the Diocese of Charleston, South Carolina, where he completed his theological studies. He was ordained to the priesthood by Bishop John England on September 24, 1825. Barry was then sent to Georgia, where he served as pastor of the Church of the Holy Trinity in Augusta from 1830 to 1854. During this time, he turned his house into a hospital (and later an orphanage) in the cholera epidemic of 1832. He was also made vicar general of Charleston and superior of the seminary in 1844, and vicar general of the Diocese of Savannah in 1853.
On January 9, 1857, Barry was appointed to succeed the late Francis Xavier Gartland as Bishop of Savannah by Pope Pius IX. He received his episcopal consecration on the following August 2 at the Baltimore Cathedral from Archbishop Francis Kenrick, with Bishops Michael Portier and John Neumann, C.SS.R., serving as co-consecrators. His poor health led him to visit Europe in July 1859, and he later died at the convent of the Brothers Hospitallers of St. John of God in Paris, France. His remains were returned to the United States, where he was buried at the Church of the Holy Trinity.