Heinlein provides more detail a little later:

The battle was on one June 1813 old style between USF Chesapeake and HMF Shannon.

The officer Heinlein refers to is William Sitgreaves Cox, with the rank of midshipman and acting fourth lieutenant on USS Chesapeake at the time. Chesapeake had a new commander, Captain James Lawrence, who replaced the previous captain due to illness. Cox was a personal friend of Lawrence, and had served under him before.

When Chesapeake sailed to fight Shannon, she did so with a green crew, recently put together. They had not had time to drill together, nor had the captain and crew had time to work together and get to know each other. Before Lawrence's arrival to take command, morale had been poor. On the morning that they sailed to fight, Lawrence helped things by paying prize money that was owed to the crew out of his own pocket.

Against Shannon's well-drilled and efficient crew, the fight went poorly. Chesapeake was out-sailed and out-fought. Of the two frigates, Chesapeake was slightly larger, with a few more guns, and a larger crew. Lawrence intended to board Shannon, but - perhaps due to an error by Chesapeake's helmsman - Shannon was able to rake the boarding party on Chesapeake's deck with grapeshot. The grapeshot and musketry killed most of the officers. Captain Lawrence was mortally wounded, and asked Cox to take him below. Cox did so, leaving the acting third lieutenant, Lieutenant George Budd, in command.

Shannon responded to the opportunity by boarding Chesapeake. During the boarding action, Budd sent a message to Cox informing him that he (Cox) was now in command; Cox was still below. The battle was short and bloody, and ended in the capture of Chesapeake. This outcome displeased the American public, and a scapegoat was sought. The scapegoat would either be Budd or Cox, and Budd made sure that it would be Cox by accusing him of desertion of his post. Cox was charged with cowardice, disobedience of orders, desertion from quarters, neglect of duty, and unofficerlike conduct. For most of this, the only witness was Budd (who had a personal interest in somebody else being the scapegoat). Cox was cleared of the charges of cowardice, disobedience, and desertion, but found guilty of neglect of duty (for going below when a boarding action was imminent) and unofficerlike conduct (for failing to kill two American sailors fleeing their posts). As Heinlein wrote, he was convicted and cashiered.

While understandable as a piece of domestic propaganda and politics, this was hardly a fair verdict, and Cox's family fought a long battle to clear his name, finally succeeding in 1952.

This little anecdote is just one of the main naval influences in Heinlein's work. Heinlein was commissioned in the US Navy in 1929, and was retired from active duty in 1934 due to tuberculosis. Another naval bit is the sword Lady Vivamus, from his novel Glory Road, which was solidly based on his naval sword: https://www.reddit.com/r/SWORDS/comments/5tk2dq/robert_heinlein_on_lady_vivamus/ Notably, the official "replica" does carry over much of the design of the naval sword: http://filmswords.com/ladyv/lady-v.htm

EDIT: As mentioned in a comment (deleted while I wrote this edit as a reply), Heinlein refers to two incidents, the second of which is the above incident. The other is from WW2, and is described below.

My reading was hasty! Indeed, you are right. While some say that Heinlein's "Second Global War" is the Napoleonic Wars, some of his other novels make it clear that it's World War II. From The Number of the Beast,

an aircraft carrier of the Second Global War had been named "Shangri La" for a land as nonexistent as "Erewhon"

So what event in WW2 does Heinlein refer to? Probably the cruiser USS San Francisco, in the early morning of 13th November, 1942, off Savo Island in the Naval Battle of Guadalcanal.

USS San Francisco was the flagship of task group Task Group 67.4. TG 67.4, commanded by Admiral Daniel J. Callaghan, consisted of two heavy cruisers (USS Portland in addition to San Francisco), 3 light cruisers, and 8 destroyers. The fighting in the night battle can be accurately described as "confused", with San Francisco firing on USS Atlanta, flagship of TG 64.2, already badly damaged by heavy Japanese gunfire and torpedoes. San Francisco's guns killed the task group commander Admiral Norman Scott, the second US admiral killed in the war (the first being Admiral Isaac C. Kidd, killed on the bridge of USS Arizona in the attack on Pearl Harbour). San Francisco's turn for severe punishment came soon, as she was engaged by the battleships Hiei and Kirishima, and also some cruisers and destroyers. Hits on the bridge killed Admiral Callaghan, the third US admiral killed in the war, and killed or wounded most of the other officers. The most senior officer left alive and fit to command was the damage control officer, Herbert Emery Schonland. Feeling that it was essential for the survival of the ship for him to act as damage control officer, he passed command of the ship - and TG 67.4 - to the communications officer, Bruce McCandless, who had been seriously wounded by the hits on the bridge, but could still command (once he recovered consciousness). As the acting TG commander, he indeed sent signals as if he were an admiral. Once voice radio communication was restored on San Francisco, command of TG 67.4 was passed on to the senior surviving officer in the TG.

After the battle, all 4 men named above, the two dead admirals as well as Schonland and McCandless, were award the Medal of Honor (thus, McCandless's actions were most strongly vindicated). McCandless's Medal of Honor citation reads

For conspicuous gallantry and exceptionally distinguished service above and beyond the call of duty as communication officer of the U.S.S. San Francisco in combat with enemy Japanese forces in the battle off Savo Island, 12-13 November 1942. In the midst of a violent night engagement, the fire of a determined and desperate enemy seriously wounded Lt. Comdr. McCandless and rendered him unconscious, killed or wounded the admiral in command, his staff, the captain of the ship, the navigator, and all other personnel on the navigating and signal bridges. Faced with the lack of superior command upon his recovery, and displaying superb initiative, he promptly assumed command of the ship and ordered her course and gunfire against an overwhelmingly powerful force. With his superiors in other vessels unaware of the loss of their admiral, and challenged by his great responsibility, Lt. Comdr. McCandless boldly continued to engage the enemy and to lead our column of following vessels to a great victory. Largely through his brilliant seamanship and great courage, the San Francisco was brought back to port, saved to fight again in the service of her country.