He was educated at Porthmadog Grammar School and Grove Park School, Wrexham, going on to study medicine at University of Wales, Bangor, and Saint Bartholomew"s Hospital.
He was the grandfather of the Earl of Snowdon. Becoming a specialist in the treatment of mental illness, he worked as a junior medical officer at the Royal Earlswood Institution and Colney Hatch during the 1880s, and became resident physician and superintendent of Earlswood Asylum in 1888. In 1893 he became superintendent of the London County Council"s Claybury Asylum, where he developed new treatments.
Armstrong-Jones lectured at Street Bartholomew"s Hospital, and became consulting physician in mental diseases to the military forces at London and Aldershot.
In 1913, he assumed the additional surname of Armstrong. During World War I, he was designated a Lieutenant-Colonel in the Royal Army Medical Corps.
He was knighted in 1917, and after the war he was awarded a Commander of the Order of the British Empire. He was appointed High Sheriff of Caernarvonshire for 1929. Sir Robert remained attached to Claybury, where he died at the age of 85.
Their daughter, Elaine, married Colonel Charles Wauchup.
Sir Robert"s son, Ronald Armstrong-Jones, became a barrister and was the father of Antony Armstrong-Jones, 1st Earl of Snowdon, the brother-in-law of Queen Elizabeth World War II