The German, who won a record seven Formula One drivers' championships and 91 Grands Prix, collected five consecutive titles with the Italian team from 2000 to 2004.
"Our champ turns 50 today. We're all with you Michael #KeepFighting," Scuderia Ferrari tweeted.
The official Formula 1 account tweeted: "We join the whole of the #F1 family in sending Michael @Schumacher our very best wishes on his 50th birthday #KeepFightingMichael #Michael50."
F1 team Mercedes have hailed Schumacher as one of the "founding fathers" for their recent success.
Hamilton last year celebrated his fifth world championship, and his fourth racing for the Silver Arrows.
"Michael is one of the founding fathers of the success we have had in the last five years," said Mercedes team principal Toto Wolff. "There is no other driver like him and his vast experience contributed tremendously in the development of our team."
"Michael has had a tremendous impact on Formula One," added Wolff. "Not only did he set an incredible record -- a record that is yet to be beaten -- but he also shaped and changed the sport forever."
"You can be sure that he is in the very best of hands and that we are doing everything humanly possible to help him," they said.
Schumacher's daughter Gina also published a short message to her Instagram page, writing: "Happy Birthday to the best dad", accompanied with three pictures.