The 33-year-old Portuguese forward said he is ready for a “new stage” in his life after agreeing to join the Italian champions in a four-year deal.
“Real Madrid would like to express its gratitude to a player that has proved to be the best in the world and has carved out one of the most brilliant eras in the history of our club and world football,” the club said.
“Real Madrid will always be his home.”
In an open letter, Ronaldo said his years in Madrid were “possibly the happiest of my life” but “the time has come for a new cycle”.
“I believe that the time has come to open a new stage in my life and that is why I have asked the club to accept transferring me,” he said.
“Real Madrid has conquered my heart, and that of my family, and that is why more than ever I want to say thank you.”
Juventus president Andrea Agnelli was at Ronaldo’s hotel in Greece on Tuesday as discussions continued over the transfer.
At the same time, Ronaldo’s agent Jorge Mendes was believed to be meeting Real Madrid.
The five-time Ballon d’Or winner joined Real Madrid from Manchester United in 2009 for a then world-record fee of £80m.
He has gone on to become the Spanish club’s all-time top scorer with 451 goals.
As well as the individual records, Ronaldo has also added two La Liga titles, two Copa del Reys and four Champions Leagues to his trophy collection during his time in Spain.
Ronaldo has been on holiday in Greece after Portugal crashed out of the World Cup against Uruguay in the last 16.
He had captained his national team at the tournament, scoring an incredible hat-trick in the country’s opening 3-3 draw against rivals Spain.