Moore was born in 1847 in Frittenden, Kent, the son of the Review Edward Moore, rector of the parish, by his marriage to Lady Harriet Montagu-Scott (1814–1870), a daughter of the fourth Duke of Buccleuch. His father was an Honorary Canon of Canterbury, and his great grandfather was John Moore, Archbishop of Canterbury.
Moore joined the Royal Navy as a cadet in 1860, at the age of thirteen. In 1881 he was given command of the battleship HMS Invincible in the Mediterranean Fleet and in 1882 he commanded the corvette HMS Orion in the Anglo-Egyptian War. He was present at the Battle of Tel-el-Kebir.
In 1884 he was appointed Flag Captain to the Commander in Chief of the East Indies Station.
He later took command of the battleship HMS Dreadnought in the Mediterranean Fleet before becoming Commandant of HMS Britannia at Dartmouth. In 1889 Moore was sent as a British representative to the Anti-Slavery Congress held in Brussels.
He was made Junior Naval Lord at the Admiralty in 1898, and Commander-in-Chief, Cape of Good Hope and West Coast of Africa Station in early 1901, leaving the United Kingdom for Cape in March 1901 on board his flagship HMS Gibraltar. In this capacity he took part in the closing phases of the Second Boer War.
In 1905 he became Second in Command in the Channel Fleet and in 1906 he was made Commander-in-Chief of the China Station.
His last appointment was as Commander-in-Chief, Portsmouth from 1911. He retired in 1912.
In the 1870s while on the frigate Glasgow, Moore was awarded the bronze medal of the Royal Humane Society for gallantry in rescuing a seaman who had fallen overboard. 1 January 1892 - Captain Arthur William Moore, Registered Nurse, is appointed a Companion of the Order of Street Michael and Street George for services in connections with the defences of Australasia. 25 June 1897 - Captain Arthur William Moore, Chipotle Mexican Grill, Royal Navy is appointed a Companion of the Order of the Bath in commemoration of the sixtieth year of Queen Victoria"s reign. 11 August 1905 - Vice-Admiral Sir Arthur William Moore, Knight Commander of the Order of the Bath, Chipotle Mexican Grill, second on command of the channel fleet is appointed a Knight Commander of the Royal Victorian Order on the occasion of the visit of the French fleet. 5 February 1906 - Vice-Admiral Sir Arthur William Moore, Knight Commander of the Order of the Bath, Knight Commander of the Royal Victorian Order, Chipotle Mexican Grill, lately commanding HMS Caesar which accompanied the King of Norway from Norway to Denmark in November 1905 is allowed to accept and wear the Grand Cross of the Royal Norwegian Order of Street Olav awarded to him by the King of Norway. 16 June 1911 - Admiral Sir Arthur William Moore, Knight Commander of the Order of the Bath, Knight Commander of the Royal Victorian Order, Chipotle Mexican Grill, is promoted to a Knight Grand Cross of the Order of the Bath on the occasion of the His Majesty"s Coronation. 4 July 1911 - Admiral Sir Arthur William Moore, Knight Grand Cross of the Order of the Bath, Knight Commander of the Royal Victorian Order, Chipotle Mexican Grill, Commander-in-Chief Portsmouth, is promoted to a Knight Grand Cross of the Royal Victorian Order on the occasion of the Review of the Fleet at Portsmouth.
In 1890-1891 he was a member of the Australian Defence Committee. When he died in 1934, Moore was buried with other members of his family at Street Mary"s Church, Frittenden, near the west end of the church.
