You might be interested in this answer by u/DBHT14 about the exact same scene, and this answer from r/warcollege which does not cite sources but is certainly still valuable. For anyone unfamiliar, here is the scene in the movie. The story it seems to be referring to is briefly mentioned in the 6th chapter of the book. Here it is:

The armada's momentum is inexorable now, fated, a force of history. Nothing will stop the surge. A man on one of the ships loses his balance, pitches over the side, and finds himself terribly alone in the Pacific Ocean. His craft does not stop. None of the ships stop. He waves his arms in panic. The ships churn past him. His horrified comrades look on as his figure recedes, then vanishes. The armada cannot stop for one man. The armada has an appointment, and means to be on time.

So it's a rather fuzzy story even in the source material, though as the r/warcollege answer notes, there are other accounts of men falling overboard (perhaps the same man falling overboard) in the same convoy.

I have no doubt that his ship would not have stopped. However, that's all the book claims. The movie seems to imply more than that with the soldier wistfully saying "So much for no man left behind." There's at least a good chance that some effort would have been made to save him.

The problem for his ship is that the troop transports are all moving in a specific formation at a specific speed and required to keep a certain distance from one another. One transport can't easily stop, then continue on, without breaking that formation, requiring the ship behind it to change course, which would require the ship to the side of it to change course, and soon the whole group is a mess and falling behind.

However, a large convoy of troop ships like this would be protected with smaller escorting vessels; the US Navy got very good at escorting its troop ships during the war. These might have been destroyers, or smaller destroyer escorts, or frigates, or other smaller auxiliaries. I can't seem to find a full list of the ships that were specifically in Task Force 56, which carried the Marines to Iwo Jima, but there were several hundred total ships involved in the invasion. The procedure likely would have been for the transport that had the man overboard to signal that it had a man overboard with whatever communication was permitted; even signal lamps could be used here. The unfortunate Marine would have had to wait a good bit and watch a number of ships pass him before one of the smaller, faster escorting vessels hopefully found him and fished him out.

The Navy had SOPs for doing this during operations that were particularly hazardous. For example, sailors had to be on exposed areas of their ships for underway refueling, which could be hairy in heavy seas. It wasn't unusual for men to be swept overboard, but ships that were linked with hoses for refueling couldn't stop either. Destroyers or other smaller ships would often sail behind the ships during these refueling operations just in case. The same often occurred during flight operations, when there was a chance of someone on the deck of an aircraft carrier being blown overboard. Procedures like these are described briefly in Ian Toll's "Pacific Crucible." He doesn't speak specifically to convoy operations, but it seems likely to me that there would be an effort to recover the man overboard -- just not by the ship he fell off of.

That's not to say that the Navy had a 100% success rate at this. Accidents happened and men died often. This article from the US Naval Institute reports that 104 men were lost overboard in the US Navy and Marine Corps in 1946; I think it's safe to assume the numbers for 1945 would have been higher.

A man falling overboard in a convoy like this almost surely would not have been rescued by the ship he fell from. The movie probably takes some liberties, however; there probably would have been an attempt to save him, and probably a pre-determined plan for which escort ship would be responsible for dropping out of formation to search and find him. In warm waters, during daytime, with people noticing he had fallen overboard, he'd have a pretty good chance of being recovered.