The former Formula One driver was reported to have heard the news of 35-year-old Lynsay Jackson's fatal heart attack just before he was due to appear live on BBC's Saturday Kitchen at the weekend.
After the show, Coulthard and his fiancée travelled to Scotland to be with his family.
"I'd like to first take this chance to thank everybody for their overwhelming support over the last two days," Couthard said.
"It means a huge amount to my family and me. It's obviously come as a huge shock to us all and we're still trying to come to terms with it all."
Jackson was married and had given birth to a daughter eight months ago.
She used to run Coulthard's fan club and the David Coulthard Museum in Twynholm.
Since retiring from Formula One in 2009 and from competitive racing in 2012, Coulthard now works for the BBC as a pundit and commentator.