As one of the finest players without a major title, Fleetwood was aiming to win the biggest individual title of his career by earning Olympic gold.
Fleetwood started the final day one off the lead as he trailed playing partners Rahm and American defending champion Xander Schauffele.
After Saturday’s third round, he said he was not “swinging freely” and lamented his lack of birdie chances.
It was a steady if not spectacular outing – but Sunday started in a different manner.
A bogey on the first was quickly wiped out by a birdie two on the second, followed by two more birdies on three and four.
The problem was everyone else – Rahm and Scheffler in particular – made even better starts.
Another bogey on the seventh threatened Fleetwood’s chances, but a run of four birdies between the eighth and 12th holes kept him in the hunt.
Rahm had surged ahead with a five-under 31 on the front nine. Then, his round unravelled after the turn.
Bogeys on 11 and 12 were followed by a very costly double on 14, allowing Fleetwood a clear run at Scheffler – who was just finishing on the last – after another birdie on 16.
Fleetwood knows how to perform at Le Golf National.
He won the 2017 French Open here before famously leading Europe to Ryder Cup glory – as one half of ‘Moliwood’ with Italian Francesco Molinari – in 2018.
However, the popular Englishman could not see the job through and faltered as Scheffler – who shot a six-under 29 on the back nine – continued his recent dominance of the men’s game.
“As a young boy taking up the game of golf, none of us had a chance to win Olympic gold so it was never on our agenda,” said Fleetwood.
“That quickly changes when you’re part of the Olympics and it feels unbelievably special.
“I know I didn’t win gold but standing on that podium with a medal was one of the most amazing moments I’ve had as a golfer.”