Schumacher's family thanked the fans in a statement on Friday evening saying they were moved to tears while French media reported that investigators were studying a camera the German had on him at the time of the accident.
A few dozen fans had assembled outside the Grenoble hospital where Schumacher has been having treatment since Sunday's accident in the resort town of Meribel, in which he hit his head on a rock when he fell while skiing off piste.
"For us Ferrari fans, we are just like brothers, we are just one," said one fan, Carlo Buschini, who raised his cap and blew kisses in the direction of Schumacher's hospital room.
"We are here to celebrate and to celebrate his battle for life," said Buschini, speaking in Italian.
Supporters of the Scuderia Ferrari Club wore their team's red colours and carried a banner reading "Schumi, All our thoughts for you and your family."
In a blog post on its website, Ferrari said Schumacher was "tackling the most important fight of his life" and sent him "double wishes" for his birthday.
Schumacher's family said in a statement, "the incredible sympathy shown today... has utterly overwhelmed us and moved us all to tears. We are deeply grateful for it and also for all the heart-warming and heartfelt wishes for Michael to get well soon, which have reached us from all over the world."
France's BFM TV channel reported that his family had given investigators a mini camera Schumacher had apparently attached to his helmet. Investigators also interviewed his son, added BFM TV who said he was with Schumacher at the time of the accident.
Schumacher is the most successful Formula One driver of all time with a record 91 race victories in a career spanning more than two decades. The German won five successive titles with Ferrari before retiring from the team at the end of 2006.
A less successful three-year comeback with Mercedes ended last year.
On Wednesday, his agent said Schumacher was in an artificial coma in a stable condition, but no update has been given since then.
Messages of support appeared on Twitter from Schumacher's former team mates on Friday.
"Happy Birthday my brother Michael! Be strong!! Be strong!! Don't give up!! We are praying for you," wrote former Ferrari driver Felipe Massa who in 2009 spent several days in an induced coma after being hit by a bouncing spring from another car.
Another of Schumacher's former team mates, Rubens Barrichello, wrote: "Keep it up, man... I know ur (sic) fighting."