The crew of K-19 didn't play football on the ice. Bigelow's movie contains a lot of inaccuracies. Nevertheless this movie scene is quite plausible. In 1963 the crew of K-181 played football on the ice at the North pole, in 1971 the crew of HMS Dreadnought did the same thing.

"Dreadnought surfaced four miles from the North Pole. The ice was so thick that only the submarine's fin broke through the ice. ... Dreadnought remained on the surface for nine hours, while the majority of the crew went out on to the ice to play football and take photographs". (James Jinks, Peter Hennessy. The Silent Deep: The Royal Navy Submarine Service Since 1945)

In 2004 seamen of USS Hampston and HMS Tireless played football (soccer) on ice.