The 32-year-old, who signed for Real from Tottenham Hotspur in 2013, was out of contract this summer.
"I write this message to say thank you to all my team mates, past and present, my managers, the backroom staff and to the fans that supported me," Bale, who scored 106 goals in 258 appearances for Real, said in a social media post.
"I arrived here nine years ago as a young man who wanted to realise my dream of playing for Real Madrid. To wear the pristine white kit... to play at the Santiago Bernabeu, to win titles and to be part of what it's so famous for, to win the Champions League."
The Welshman will be best remembered for scoring twice, including a stunning overhead kick, in the 2018 Champions League final against Liverpool and also netting a superb winner against Barcelona in the 2014 Copa del Rey final.
Despite his fine collection of titles - three LaLiga triumphs and five Champions League crowns - Bale's time at Real was marred by injuries.
He was also often a target of the Spanish media, who labelled him a "parasite" after missing Real's 4-0 home loss to Barcelona in March, and he was jeered by some fans during a league game in April.
"To be a part of this club's history and to achieve what we achieved while I was a Real Madrid player, has been an incredible experience and one I will never forget," Bale added.
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