Last few minutes of the movie - Spoiler Alert

You didn't miss anything. The movie ends by fading out on that banner blowing in the breeze, reading "There is still time..brother". The movie goes to black at that point. There is no "The End" in the film. It just stops like that. (The "ghost town" is the city of Melbourne, where most of the action takes place; it's now deserted because everyone is dead.)

Several years ago some idiot technician somewhere added the words "The End" to a print of the movie, just to make sure everyone knew it had ended. This completely ruined the impact of director Stanley Kramer's original fade-to-black with no unnecessary "The End" attached. Fortunately, no one uses that stupid print anymore; the new DVD and Blu-ray releases don't have it. This restores the film to what it was.

And it also means you didn't miss a thing, unless literally the last couple of seconds -- of the fade to black over dramatic music -- got cut off. I hope you saw that, but even if not, you didn't miss anything substantive.

What happens after Peck's character leaves Gardner and the sub descends under the water?

As Hob said, you didn't miss anything. The story essentially ends with the end of the main characters. Dwight takes the sub out on one last Quixotic voyage back to the 'States - in the book he sensibly takes the sub out beyond the national waters limit to "sink her". I presume this means he will put the sub on the ocean floor, where he and his minimal crew will take the death pills. Moira probably in the film and certainly in the book, takes the pill along with a final swig of booze. So that's where the story ends, followed only by a brief epilogue showing that Melbourne, too, is now dead.