Atkins died at New York Weill-Cornell Medical Center and was surrounded by his wife and close friends, said Richard Rothstein, his spokesman.
Atkins suffered a severe head injury April 8, after slipping on an icy sidewalk yards from his Atkins Center for Complementary Medicine in midtown Manhattan.
Atkins underwent surgery to remove a blood clot that formed after his fall, but he remained in critical condition at New York Weill Cornell Medical Center.
Atkins first advocated his unorthodox weight-loss plan — which emphasizes meat, eggs and cheese and discourages bread, rice and fruit — in his 1972 book, "Dr. Atkins' Diet Revolution."
Its publication came at a time when the medical establishment was encouraging a low-fat, high-carbohydrate diet. The following year, the American Medical Association dismissed Atkins' diet as nutritional folly and Congress summoned him to Capitol Hill to defend the plan.
Labeling it "potentially dangerous," the AMA said the diet's scientific underpinning was "naive" and "biochemically incorrect." It scolded the book's publishers for promoting "bizarre concepts of nutrition and dieting."
As CBS News Correspondent Mika Brzezinski reports, Dr. Atkins lived to see many of his critics eat their words. Though he was long dismissed by the medical establishment, millions of people bought into his prescription for losing weight.
Atkins' philosophy enjoyed a resurgence in the 1990s with "Dr. Atkins' New Diet Revolution," which sold more than 10 million copies worldwide and spent five years on The New York Times best-seller list.
But criticism of the diet lingered, with many arguing that it could affect kidney function, raise cholesterol levels and deprive the dieter of important nutrients.
Atkins said no study showed that people with normal kidney function developed problems because of a high-protein diet, and he never gave in to his detractors.
"He raised the level of awareness of the importance of reducing the intake of simple carbohydrates in the diet, which we agreed with," Dr. Dean Ornish, the Clinical Professor of Medicine at the University of California, told CBS News. "Although our recommendations were very different, I never doubted his sincere belief in what he advocated."
This year, his approach was vindicated in part by the very medical community that scorned him. In February, some half-dozen studies showed that people on the Atkins diet lost weight without compromising their health. The studies showed that Atkins dieters' cardiovascular risk factors and overall cholesterol profiles changed for the better.
Still, many of the researchers were reluctant to recommend the Atkins diet, saying a large new study now under way could settle lingering questions of its long-term effects.
On the Atkins diet, up to two-thirds of calories may come from fat — more than double the usual recommendation, and violating what medical professionals have long believed about healthy eating. Carbohydrates are the foundation of a good diet, most say. Eating calorie-dense fat is what makes people fat, they say, and eating saturated fat is dangerous.
To Atkins, the key dietary villain in obesity was carbohydrates. He argued they make susceptible people pump out too much insulin, which in turn encourages them to put on fat.
"If you cut out carbohydrates, you automatically burn stored fat. That's the basic principle. It's been known about for 35 years," Atkins told CBS News Correspondent Elizabeth Kaledin in a 2001 interview.
Fat in foods can be a dieter's friend, Atkins said, in part because it quenches appetite and stops carbohydrate craving.
Atkins, a graduate of Cornell University's medical school, first tried a low-carbohydrate diet in 1963 after reading about one in the Journal of the American Medical Association. He said he lost weight so easily that he converted his fledgling Manhattan cardiology practice into an obesity clinic.
Besides his work on nutrition, Atkins also argued that ozone gas can kill cancer cells and HIV, the virus that causes AIDS, and he claimed to have treated more than 1,000 patients with ozone therapy.
The ozone treatment is a common alternative therapy in Germany and some other nations but has not gained acceptance in the United States.
In 1999, Atkins established the Robert C. Atkins Foundation to finance diet research. It has sponsored research at Duke University, the University of Connecticut and Harvard.
Last April, Atkins was hospitalized for cardiac arrest, which he said was related to an infection of the heart and was not related to the diet.
Besides his wife, Veronica, Atkins is survived by his mother, Norma, of Palm Beach, Fla.