Colin Weir and wife Chris, from Largs in North Ayrshire, claimed the £161 million prize in July 2011.
Earlier this year the pair confirmed they would divorce after 38 years of marriage.
"We would ask for privacy for his family and friends at this distressing time.
"No further comment will be made other than to offer sincere thanks to the staff of University Hospital Ayr for their care and compassion."
Scotland's First Minister Nicola Sturgeon said she was "incredibly sad" to hear of the death.
She said: "Colin's determination and generosity in the cause of Scottish independence cannot be overstated and was hugely appreciated.
"The SNP and the independence movement has lost a true friend today and we will miss him dearly."
The 71-year-old former TV cameraman and his then-wife, a former psychiatric nurse, made the Sunday Times Rich List with their win eight years ago.
News of their intention to split broke in April this year with outlets reporting it had been granted in September.
Together they have two grown-up children.
In 2013 the couple set up The Weir Charitable Trust and made a donation to a community football club in Largs.
They also invested in Partick Thistle Football Club which led to the youth set-up being rebranded the Thistle Weir Youth Academy and a section of their Firhill Stadium being named the Colin Weir Stand.
In November of this year Mr Weir secured a majority shareholding at the club and promised to give the 55% shareholding directly to a fans group by March 2020.
The club tweeted: "It is with deep sadness that we confirm that lifelong Jags fan Colin Weir passed away earlier today.
"On behalf of everyone at Partick Thistle, our love, thoughts and prayers are with the family and close friends of Colin at this most difficult time."
The Weirs also defended making a donation of £1 million to the independence campaign ahead of the 2014 referendum, and continued donating to the SNP afterwards.
SNP MSP Mike Russell, the Scottish Government's Cabinet Secretary for Government Business and Constitutional Relations, also tweeted about the news.
He said: "Sad to see this - remember him very kindly, both of us contesting seats in 1987, working on PPBs for @theSNP when I was VC Publicity & his great generosity after his win."
Their £161 million prize was the UK's biggest win until October this year, when an anonymous ticket holder claimed a £170 million jackpot.