He was a leading advocate of the value of nautical surveying in relation to naval operations. His early career included service under Robert FitzRoy on the HMS Beagle during Charles Darwin's voyage of 1836, and Bartolomé Island in the Galapagos Islands was named after him. From 1842 to 1846 he commanded HMS Philomel on the South American Station and surveyed the Falkland Islands. During the Crimean War he was sent by Sir Francis Beaufort, Hydrographer of the Navy, to the Baltic to assist the fleet commanded by Sir Charles Napier. Sulivan, commanding the paddle steamer HMS Lightning, made many invaluable surveys and charts of the shallow waters in which the fleet had to operate, and led the bombardment ships into position during the capture of Bomarsund. From 1856 to 1865 he was the naval professional member of the Board of Trade. He was promoted to Vice-Admiral in 1870, and Admiral in 1877. After Robert FitzRoy committed suicide in 1865, leaving his wife and daughter destitute, Sulivan convinced the British government to provide them with £3000, to which Charles Darwin contributed another £100 of his own money.
He was created a CB in July 1855, and in June 1869 a KCB.