The family, which has been periodically posting messages on Schumacher's website, wrote Friday night that they were "overwhelmed" by the show of support.
"The incredible sympathies shown today by the Ferrari Fans outside the hospital has utterly overwhelmed us and moved us all to tears," it said.
The seven-time F1 champion has undergone two brain operations since the accident. Doctors are keeping him sedated and his body at an artificially low temperature to help with the healing. The physicians have refused to give a prognosis for Schumacher, saying they are focused on his immediate care.
Schumacher, known affectionately as Schumi by fans, is the most successful F1 driver in history, racking up a record 91 race wins. He retired in 2012.
Ferrari on Friday sent special birthday greetings to Schumacher, recalling the 72 victories he won from 1996-2006.
"Today is Michel Schumacher's birthday. In these hours he's on the most important race of his life and we want to dedicate a special honor to him," the statement said. "Forza (Come on) Michael!"
Associated Press writers Nicole Winfield in Rome and Lori Hinnant in Paris contributed to this report.