Dutton, a six-time Olympian, twice took team gold representing Australia before he became an American citizen in 2006 and began competing for the country he has lived in since 1991.
"In Australia people learn to ride not quite so much in an orchestrated way... they learn to ride on their own a bit more," said Dutton, who grew up riding on his parents' sheep and wheat farm. "In America it's a bit more managed.
"The U.S. has had some great riders as well traditionally, hopefully there will be more medals ahead."
France wins equestrian team eventing gold while Jung takes individual for Germany
Eventing, one of three Olympic equestrian competitions, requires galloping over giant obstacles in an open cross country field and also completing a dressage test and stadium jumping.
Dutton needed to turn to sheer grit as well as skill to get through Monday's tough cross country course when he lost a stirrup but still managed to clear a difficult combination with the horse Mighty Nice.
The 2004 Irish Sport Horse had a personal best in dressage, and was in fourth place with Dutton, who is based in Pennsylvania, going into Tuesday's final show-jumping round.
"I had a rail down and was thinking hopefully I'd stay in fourth, then the guy in front of me had two rails down," said Dutton, who was seen hugging Mighty Nice after the competition.
Of four Americans, only Dutton and Boyd Martin, who was also born in Australia, finished cross country. That put team USA out of medal contention.
The team event was won by France, with Germany second, whose team member Michael Jung nabbed his second consecutive individual gold with the horse Sam.
Dutton's former team, Australia came third.
Turning 53 next month, Dutton is older than most Olympic athletes, but in equestrian, where men and women of a wide age range compete together, he was not the most senior on the field. New Zealand's Mark Todd competed in his seventh Olympics at 60.
"I try to watch what I eat, depending on what the day is like I do some other sports," Dutton said of his fitness regime.
"He (Mighty Nice) may have another Olympics," he added.
Dutton previously visited Brazil in 2007 for the Pan American Games, also held in Rio's Deodoro area.
"The Brazilian people have tried really hard. It's not easy for them, a lot of other things need money to be spent on as well," he said. "But I can't be more pleased with how welcoming they have been."
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