LRCPI and LM, LRCSI and LM 1956; MSc Manch 1961, PhD 1967; FFOM RCPI 1976.
Conroy was educated at Royal College of Surgeons in Ireland and held house appointments in the Richmond Hospital, Dublin. After a period in general practice, Conroy entered the University of Manchester as a research student in physiology and took an MSc. In 1961 he returned to Dublin as medical registrar at the Richmond Hospital and assistant in pharmacology at the Royal College of Surgeons. He returned to Manchester in 1963 where he joined the staff of the department of physiology and took his PhD. He returned again to the Royal College of Surgeons in 1969 as reader in physiology, being appointed professor in 1970.