Coulthard's sister, Lynsay Jackson, was found dead in her home in the village of Crossmichael, Kirkcudbrightshire, on Saturday morning.
Reports suggest that the 35-year-old died from a suspected heart attack after cardiac failure at around 4am.
It is understood she went to bed complaining of stomach pains before the attack.
Coulthard said: "I'd like to first take this chance to thank everybody for their overwhelming support over the last two days. It means a huge amount to my family and I. It's obviously come as a huge shock to us all and we're still trying to come to terms with it all."
Police have confirmed they are not treating the death as suspicious. However, tests are still being conducted to try to establish what caused the tragedy.
A Dumfries and Galloway Police spokesman said: "We can confirm a 35-year-old woman was found dead in her home in Crossmichael. Our inquiries into her death are ongoing, but there are not believed to be any suspicious circumstances and a report is being prepared for the procurator fiscal."
Former F1 driver Coulthard, 41, was told the news about his younger sister just moments before he was due to appear on the BBC show, Saturday Kitchen Live. He was given the option of dropping out of the show by producers, but instead decided to appear on the 90-minute programme hosted by TV-chef James Martin.
Coulthard's agent James Williamson said: "David heard just before he went on air. But amid the shock, he wanted to put on a brave face and just get the show done."
Mrs Jackson, who married environmental health officer Will, 34, in 2011, had an eight-month old daughter. Trained as a nurse, she previously ran the David Coulthard Museum in Twynholm, as well as his fan club.
Her parents, Joyce, 65, and Duncan, 67, from Twynholm, are believed to be flying back home to the UK from a holiday in Australia.
Shortly after the news broke, tributes flooded in to the family via social networking site Twitter.
Coulthard's ex-BBC colleague Jake Humphrey said he had spoken to the racer and he was "thankful for our thoughts at this difficult time".
F1 commentator Jonathan Legard said: "Another tragic weekend shock – news that @therealdcf1 sister has been found dead at her home in Scotland. Thoughts are with all family."
Racing group McLaren, whose cars Coulthard drove, tweeted: "We offer our sincere sympathy, on behalf of all at McLaren, to the Coulthard family at this dreadful time."
David Coulthard, known as DC, dominated Britain's junior Formula Ford categories in 1989 and became the first recipient of the McLaren Autosport BRDC Award for young drivers. He made his Formula One debut in 1994 and won 13 Grand Prix before his retirement from the sport several years ago.
He had notched up 535 points, making him the highest-scoring British driver of all time. Since his retirement he has regularly appeared as a pundit on the BBC's F1 coverage. He has lived for many years in Monaco, but also has homes in several European countries and owns a number of luxury hotels.