Sir David, one of Scotland's richest businessmen, said the country's record on obesity and heart disease were as much a cause of shame as sectarianism.
And he said massive investment in sport, including a dedicated national institute, was needed to tackle the country's poor health record and elevate Scottish sport's world standing.
Sir David said: "I've just gone through a major thing with what was called 'Scotland's shame' on sectarianism at Rangers and we've worked very hard to eliminate that.
"There has been a vast improvement, and all credit must be given to the fans.
"Scotland's shame is obesity, drugs and a lack of opportunities for kids in sport. We've got to encourage and nurture them and give them the facilities."
The Rangers chairman made his comments at Murrayfield, home of Scottish rugby, as his firm, Murray International, was unveiled as the national team's new sponsor.
The 2.7 million, three-year deal, will replace The Famous Grouse, which steps down after 17 years as the main sponsor.
Sir David said: "It's crucial that business gets behind Scottish sport. I'm very disappointed when I see so many companies not aiding sport.
"We, as a nation, have fallen behind many other countries in sport and it's important the sort of money we're putting into the game hopefully encourages other people not only to help rugby, but all sport." He added: "When you run a big business in Scotland, you have a social responsibility to support sport, and I hope other people take the lead. Funding is crucial to the development of indoor sporting arenas throughout the country, and I think we fall badly down in that area."
Asked if he felt the government was investing enough in sport, Sir David replied: "Why have we not got a national institute of sport today, where the elite golfers, athletes, swimmers, football players and rugby [can train]?
"We need a facility where the finest athletes can be encouraged, and train with some of the best coaches in the world, who can come here and then train Scottish coaches to take over.
"Look at the national institute of sport in Australia - we have nothing in Scotland.
"I don't think there's a policy or any leadership in sport. Why is it our best athletes have to leave this country to find the best coaching?"
Scotland's health record is one of the poorest in Europe. Scotland has the highest level of obesity in the UK, with the number of obese children double the UK average. Scotland also has a rate of deaths from cancer 15 per cent higher than the UK average.
A spokeswoman for the British Medical Association Scotland welcomed Sir David's comments and said she hoped they would inspire more business leaders to get involved in tackling health issues affecting their staff and communities.
She added: "Promoting a healthy lifestyle should be congratulated. Physical inactivity is one of the biggest contributors to coronary heart disease, which effects 500,000 Scots each year.
"It is important to encourage people to have healthier, more sporty lives."