The Londoner, 21, started the combined running and shooting event in the silver medal position, but ended up finishing in 10th place in Rio.
Russia's Aleksander Lesun, 28, streaked ahead of the field to win gold.
Ukraine's Pavlo Tymoshchenko took silver ahead of Mexico's Ismael Marcelo Hernandez Uscanga.
Jamie Cooke, who was thought to GB's best hope of a medal, finished a disappointing 14th.
Britain have won five medals in the women's event since it was introduced to the Olympics in Sydney 16 years ago.
Choong was a surprise medal contender going into the final round, having rarely challenged for the podium in any of his previous international outings.
He secured a spot for the Games by finishing seventh at the 2015 European Championships.
Despite his lack of top-level pedigree, he performed impressively in the fencing, swimming and horse-riding rounds to start nine seconds behind Lesun in the combined event.
Choong made a promising start, holding his position after two of the four rounds, which see athletes run 800m before stopping at the shooting range.
But he lost a lot of ground in the third shooting round, struggling to hit his five targets as a host of athletes overtook him.
"Everything went really well except for the shooting," he said.
"I was very happy with my first shoot and the last was the same but the middle I don't know what happened."
Stephanie Cook, 2000 Olympic gold medallist:
"This was an amazing position for Joe to be in at the start of the combined event.
"He gave it his all out there and should be proud of what he has achieved. He has gleaned so much experience from this competition.
"Unfortunately, on this occasion, there were better, more experienced athletes out there. It was a real battle for those medals."
More to follow.
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Joe Choong failed to become Britain's first Olympic medallist in the men's modern pentathlon after slipping out of contention in the final round.