The baby, who was being fed with gastric and intravenous tubes, was snatched on Friday evening from a hospital in the city of Toulouse.
"His life is at risk if he does not receive medical care immediately," prosecutors said in a statement.
The 33-year-old father was separated from the boy's mother but they appeared to have been on cordial terms and he visited the baby in hospital every day.
"We don't really have much information," French news agency AFP reports, adding that the kidnapping was "incomprehensible."
A hospital official said: "The only thing that matters for us is to find the baby really fast."