Madrid, fifth in La Liga, announced the signing of the Spanish midfielder for a reported 15 million euros (£13.4 million) from the Premier League club after a 2-0 defeat by Real Sociedad on Sunday.
"This is the happiest day of my life, I couldn't have a better gift for Three Kings Day (Jan. 6 in Spain)," Diaz, 19, told reporters.
"I had three options after deciding to leave Manchester City. One, play at Real Madrid, two, play at Real Madrid and three, play at Real Madrid. It was impossible to go anywhere else.
"I understand the responsibility of playing with this shirt. I will give everything from the first day."
Diaz signed a six-and-a-half year deal which will keep him at the Santiago Bernabeu until 2025.
Madrid president Florentino Perez praised the youngster and said his enthusiasm to join the European champions will help him succeed.
"This player has a special talent. He has quality, magic and dreams of succeeding at this club. He chose it above others and that energy will help (him)," Perez said.
(Reporting by Rik Sharma; Editing by Christian Radnedge)
