Lieutenant Colonel Lewis "Chesty" Puller saluting Basilone when he gets the Medal of Honor (Basilone said in real life that it was one of the proudest moments of his life)
Normally, the lower-ranking man salutes the higher-ranking one, and holds the salute until the superior returns it. However, any man wearing the Medal of Honor gets saluted by everybody, from the lowest private to five-star General. Strictly speaking, they are saluting the medal and the act that earned it.
Runner calling Leckie Peaches after they're both injured and brought on the same hospital ship. Also when Leckie starts to get choked up while explaining that he meant to come back for Runner, but couldn't, and Runner tells him that he understands.
Leckie: (tearing up) I tried to get a Corpsman. I tried to get back to you.
Runner: I know. Lucky? (Leckie looks at him) I know.
Captain Ack-Ack admitting that he's comforted by the thought that his father, who manages a textile mill that makes fabric for the Military, had something to do with the blankets he and his men sleep in. Also makes the moment when his dead body is wrapped in a blanket that much more if a Tear Jerker
For all that Snafu makes fun of Peck, it is rather touching seeing him comfort Peck after his meltdown.
When the men return from the Guadalcanal campaign, they wonder whether what they did was worth it. Then when they get coffee from a cook, the cook informs them that their exploits are front page news all over America and everyone back home considers them heroes.
A Real Life example: Sledge's family was so impressed by John Mazzello's portrayal of Gene, that they gave him the pipe Sledge carried during the war.