He attended Wesley College, Dublin and Trinity College, Dublin as well as studying in Vienna, before graduating in medicine in 1889. He became President of the Royal College of Surgeons in Ireland 1910-11. He was Professor of Laryngology and Otology at Trinity College.
He was MP for Dublin University 1918-1922, having previously been defeated in a 1917 by-election for the same constituency.
Woods left the House of Commons at the dissolution of 1922 when his constituency ceased to be represented in the House of Commons.