I did not see the weapon in the video, (which is more of a hippy anti-war propaganda piece than a true documentary in my opinion).
From your description, I think you may have seen the "Weapon Alfa [or Able or Alpha] which was a rocket thrown depth charge for anti-submarine use. You can Google the term for videos showing it in operation. It was a big improvement over conventional depth charges rolled off rails or even launched from “Y” or “K” guns which required the destroyer to basically drive directly over the submarine’s location. With Weapon Alfa you could stand off a distance and since the launcher was trainable, not fixed, you could fire at any point from dead ahead to about 100 degrees back on either side of the ship.
Weapon Alfa was replaced by the ASROC, Anti Submarine Rocket Launcher which was an eight cell box type launcher. These held a single rocket in each cell, with a range of up to 24,000 yards, and the warhead was either a Mark 44 or 46 acoustic homing torpedo or a nuclear depth charge.