Palmer was commissioned lieutenant on 23 April 1778 and promoted commander on 1 December 1780. Following the unsuccessful Battle of Chesapeake Bay on 5 September 1781 his command, the fireship Vulcan, was marooned in the York River, and so he sought permission from his senior officer, Captain Thomas Symonds, to attack two French sail of the line and a frigate. Unfortunately four craft that joined him in the attack were precipitately set alight and the French were able to cut their cables and make sail. Alone at the wheel, Palmer tried to grapple one of the Frenchmen before making good his own escape in a boat, only to see the blazing Vulcan drift harmlessly ashore. On 15 June 1782 he was honourably acquitted of any blame for the loss of the Vulcan at a court-martial aboard the Warspite at Portsmouth.
He was posted captain on 18 January 1783, and in July recommissioned the Perseus 20 for service out of Liverpool and off Ireland. This vessel was refitted at Plymouth in 1784, and after further service off Ireland she was paid off in February 1787. In December 1791 he recommissioned the Perseus 20, going out to Jamaica in March 1792, and at the commencement of hostilities with France in 1793 he was commanding her in the Leeward Islands from where he returned almost immediately to England. She was paid off in December 1794.
Palmer recommissioned the Lion 64 in February 1795, and in December exchanged with Captain Edmund Crawley into the Adventure 44, which he commanded only briefly.
He did not thereafter see any further employment but was promoted rear-admiral on 23 April 1804, vice-admiral on 31 July 1810, and admiral on 12 August 1819. He died on 8 September 1834 at his residence near Esher, Surrey.
In 1786 at Liverpool Palmer married Mary, the daughter of the future Rear-Admiral Richard Smith. He was thus related through marriage to Admiral Alexander Wilmot Schomberg.