Henry was the second of three sons, born to a prominent Adelaide doctor Harold Rischbieth who was an honorary surgeon at the Adelaide Children's Hospital and his wife Eileen May (née Beresford). Henry's early education was at St Peter's College, an association which he maintained in later life, when he became the school doctor. Following the completion of his medical course at the University of Adelaide he served with the Royal Australian Navy, his first overseas posting being to HMAS Tarangan, the naval base at Milne Bay. Subsequently he served with distinction on HMAS Warramunga, ending his service in 1946 with the rank of lieutenant-commander.

He returned to South Australia, and after a year as RMO at the Adelaide Children's Hospital, he was appointed medical superintendent. Having obtained his MRACP in 1950, he undertook further postgraduate studies at Great Ormond Street, London. He subsequently entered private practice in Adelaide as a paediatrician with attachment to the Adelaide Children's Hospital, culminating in the position of senior visiting physician at the time of his death.

Henry's private practice was extensive, but he always found time for wise counsel to the parents of his young patients, and he showed remarkable recall of names of patients and their ancestral medical histories. He was particularly adept in the management of the new-born and later in his career was widely respected for his management of learning difficulties in school children. His contribution to the Adelaide Children's Hospital was impressive, both at the clinical level and administratively. He was very popular with the medical undergraduates, with his teaching rounds being enlivened with many clinical examples gathered from his wide experience in paediatrics.

Henry was widely involved in many other activities. He had been a sidesman at Christ Church in North Adelaide for many years, and was its representative on Synod. He was a staunch supporter of Legacy and always found the time to support the families allotted to his care. He was a keen tennis player, and he took to bowls in later years with skill and enterprise. Henry had a very happy family life with his wife Nancy and four children supporting him throughout his busy professional life. One of his daughters is a social worker, and two other daughters have graduated through the Adelaide Children's Hospital College of Nursing. His younger brother, Richard Henry, is a well respected neurologist in Adelaide.