Yes, the British did undertake several aggressive actions against the French Navy immediately after the Fall of France. Ships in British harbours were seized, including two older battleships, Paris and Courbet. This was mostly peaceful, though a firefight did occur aboard the large submarine Surcouf, leaving four dead. The Smithsonian programme was likely referring to the attack on Mers-el-Kebir, the Algerian port at which a large portion of the French Navy's most modern ships were based. The Royal Navy sent its Force H, with several battleships and the aircraft carrier Ark Royal to either welcome these ships to the Free French Navy, escort them to French Caribbean territories out of reach of the Axis powers, or sink them if necessary. Negotiations between the British and French commanders broke down. The British responded with a combined air and naval attack, sinking most of the ships in the harbour. A few days later, aircraft from HMS Hermes attacked the French battleship Richelieu at Dakar, causing heavy damage to her. The Vichy French response was limited to ineffective air raids on Gibraltar.

There would be several further clashes between Allied and Vichy forces. In September 1940, the British and Free French made an abortive attempt at capturing Dakar. This would see the battleship HMS Resolution crippled by a Vichy submarine, while the British sank two submarines and heavily damaged a destroyer. No troops would be landed, due to heavy shore defenses. In June 1941, forces from the British Empire, aided by the Free French, would invade Syria and the Lebanon, both of which were under Vichy French control. Fighting here was fierce, but the Vichy forces could only hold out for a month, as they were poorly supplied and equipped, and facing overwhelming force. Following Japan's entry to the war, the British invaded the French colonies in the Indian Ocean. This was done to deny the Japanese the use of Madagascar and Reunion as submarine bases. This was relatively uneventful, though HMS Ramillies would be heavily damaged by a Japanese midget submarine. Finally, Anglo-American forces would invade Vichy-held North Africa as part of Operation Torch. Vichy naval forces based at Casablanca opposed the landings there. The incomplete battleship Jean Bart was used as a shore battery, and a cruiser and destroyer force sortied against the landing ships. Four destroyers and a light cruiser were sunk, for negligible loss to their American opponents. Once the troops were ashore, against heavy opposition in some areas, the Vichy administration in North Africa was swiftly toppled by a coup, and resistance ceased. Torch was the trigger for the Germans to dismantle the Vichy regime, and hence there would be no further combat between Allied and Vichy French forces.