My dad, a naval veteran, choked up a bit at that scene. You're not the only one—it's far and away the best scene of the movie.
Even though the younger soldiers were consistently surly, wisecracking and sarcastic smartasses throughout the film, the sheer warmth, affection and respect that they treat said old-warriors with resembles the type little-kids have for their favorite grandpas, and is very endearing to watch.
There is also the very nice moment when a vet speaks to Cora Raikes and welcomes her on duty, not as a woman in "his navy," but as a destroyer crew member ready to perform with a battleship.
In the novelization, it goes into detail about why the veterans agreed to help Stone by saying they had seen the horrors of war and vowed "never again."
In addition to the veterans that have served on the Missouri, some of the vets' caps shows that they have served on other ships, one in particular the USS North Carolina.
Also watching Canales who has literally admitted that he believes himself to be half a man and has "no will to live" slowly regain his posture and decorum as a warrior and then go head to head with an advanced alien as a double amputee. Doubles as a Crowning Moment of Heartwarming and a Crowning Moment of Awesome. By far, my favorite character and plotline of the entire movie. Watching him get his medal first choked me up a bit.
This movie achieved a trifecta of heartwarming tributes: to elderly retired veterans, to disabled veterans of modern wars, and to the spirit of international cooperation. Impressive!