Simpson, who won gold in the Star class at the Beijing Olympics in 2008, died when he became trapped as the 72-foot catamaran the Artemis, Sweden's entry in the America's Cup, overturned.
"We're devastated by the news from San Francisco," RYA performance director John Derbyshire said in a statement from the Royal Yacht Association and British Sailing Team.
"Andrew is someone I've worked closely with since the age of 16 - he was a great talent, and a key figure in our world class programme over many years culminating in his well-deserved Olympic success.
"He was a huge inspiration to others, both within the British Sailing Team and across the nation and our deepest sympathies go out to his family at this terrible time."
Simpson, nicknamed 'Bart' after the cartoon character, also won a silver medal at the London Olympics last year, again in the Star class, and was a close friend of four-times Olympic champion Ben Ainslie.
A statement from Ainslie's Oracle team, which was also testing its AC72 on San Francisco Bay read: "Today is a sad day for all of us in the sailing community.
"Andrew Simpson was a great person, a terrific sailor, and a good friend to all of our team.
"Our thoughts are with his family and the entire Artemis team. He will be dearly missed."
The RYA's Olympic team manager Stephen Park said Simpson was a very popular member of the team.
"Andrew was a fantastic sailor who got the best out of everyone he sailed with," he said. "He was much loved."
The British Olympic Association said it was "devastated" at the tragic death of Simpson.
"Andrew was a treasured and accomplished member of Team GB, both at the home Olympics in London 2012, where he won silver with childhood friend Iain Percy in the Star class, and at the Beijing 2008 Olympic Games, where he won the gold medal also sailing alongside Iain," a BOA statement read.
"Andrews's talent and humour was an inspiration to others and he will be sorely missed by the Olympic Family."
His death also touched Olympians in other sports, including Spanish tennis player Rafa Nadal.
"A tragedy what happened in the Americas Cup," he said on Twitter. "All my support to the family of the great Andrew Simpson."
Simpson's twin-hulled boat was performing a so-called "bear-away" manoeuvre, turning away from the wind, when one bow dropped under the water's surface, and the vessel flipped over, America's Cup spokesman Tim Jeffery told Reuters.
Simpson, who was working as the team's tactician, ended up trapped beneath the boat in the water for around 10 minutes and had to be pulled out by rescue divers, who tried and failed to revive him.
The incident was believed to be the first fatality in connection with the America's Cup since the early 1990s, when a crew member from a Spanish team died in a training accident off the coast of Mallorca in the Mediterranean.