The retired seven-time world champion slammed his head against a rock on Sunday while skiing in the French Alps and has had two operations to remove bleeding and pressure on his brain. (Agencies)
Prosecutors have opened a probe into the accident, as is common practice in France in such cases, and are exploring the theory that the German was skiing at great speed when he fell.
Schumacher's plight has prompted an outpouring of sympathy from fans and in a message coinciding with his birthday, his family said they were touched by the reaction.
"Following Michael's skiing accident, we would like to thank the people from all around the world who have expressed their sympathy and sent their best wishes for his recovery," the family said in a statement posted on Schumacher's website.
"We all know he is a fighter and will not give up" they added.
There have been conflicting statements about the speed Schumacher was going at the time of his accident at the Meribel ski resort, where he has a property.
The impact split the helmet he was wearing in two, according to a source close to the investigation.
The Ferrari F1 team, with whom Schumacher spent many years, announced it would on Friday hold a "silent gathering" in front of the Grenoble hospital where he is being treated to mark the birthday of their ex-driver, who is being kept in an induced coma.
Schumacher's fan club in his childhood town of Kerpen said any celebration of the birthday would be in bad taste.
Jean Todt, former head of the Ferrari team, visited Schumacher's bedside on Thursday. Schumacher's wife Corinna, their two teenage children, and his father and brother were also there.
The hospital and Schumacher's manager had briefed the press daily since the accident. But they did not do so on Thursday, instead promising to communicate only if there was something new to report.
His media representative, Sabine Kehm, said Wednesday that Schumacher was stable though still critical.
The retired seven-time world champion slammed his head against a rock on Sunday while skiing in the French Alps and has had two operations to remove bleeding and pressure on his brain.