The 22-year-old has signed a contract until June 2031 and will link up with manager Álvaro Arbeloa at Craven Cottage.
Real Madrid will receive €40 million fixed, plus €2 million in add-ons. The Spanish club retain a 30% sell-on clause and matching rights on any future proposals for the player.
García, a product of La Fábrica, leaves Madrid after making appearances for the first team, scoring goals and providing assists. The move reunites him with Arbeloa, who previously coached him in the Real Madrid youth setup.