The 36-year-old was part of a 11-man crew aboard Artemis Racing's AC72 catamaran, which capsized in San Francisco Bay during training.
The sailor, known by many as Bart, was trapped underneath the boat and, despite attempts by doctors afloat and subsequently ashore, he was unable to be saved.
He was trapped under the platform for an estimated 10 minutes.
Simpson represented Great Britain at two Olympic Games, sailing in the Star class alongside Iain Percy.
The pair took gold at the Beijing 2008 Games and last year were close to topping the podium once more, only to be bumped down to silver in the medal race.
Before last summer's Olympics, Simpson admitted that winning the America's Cup was his biggest goal.
He had hoped that dream would come true as strategist for challenger Artemis Racing, whose sailing team director and tactician is close friend Percy.
San Francisco Fire Department spokeswoman Mindy Talmadge confirmed to Press Association Sport that Simpson was taken to St Francis Yacht Club on the shore where paramedics performed CPR, but was pronounced dead. She said another sailor suffered minor injuries.
"The entire Artemis Racing team is devastated by what happened," Artemis Racing chief executive officer Paul Cayard told their official website, artemis-racing.americascup.com.
"Our heartfelt condolences are with Andrew's wife and family."
Tributes have already begun pouring in for the popular two-time Olympic medallist.
Olympic silver medallist windsurfer Nick Dempsey said via his Twitter account "Going to miss you Bart", while Luke Patience, who took silver in the men's 470 class at London 2012, said: "Devastated. Bart, you are a true inspiration."
The British Sonar team, comprising John Robertson, Hannah Stodel and Steve Thomas, also took to the social network, posting: "Devastated by the news from Artemis racing... Our thoughts and prayers go out to Bart's family and team, a true sailing legend..."
Sir Clive Woodward, British Olympic Association director of sport at London 2012, was another to pay tribute: "Totally shocked to hear Olympic star Andrew 'Bart' Simpson died after Swedish America's cup catamaran capsizes - thoughts with his family."