Benitez, who began his coaching career in Real's youth ranks and replaced Carlo Ancelotti in the hot seat in June, was dismissed on Monday after his team failed to find the level of consistency demanded by president Florentino Perez.
There were also widespread media reports that the coach did not get on with some of the players while disgruntled fans have been whistling Benitez at the Bernabeu in recent months.
"I would like to use this article to say goodbye to everyone at every level of the club," he said on his website (www.rafabenitez.com) on Tuesday.
"I want everyone at the club from the board of directors, executives, workers and all of the fans to know it has been an honour and privilege to be in charge at this club where I grew up as a person, player and coach.
"As a 'madridista', steeped in the traditions and values of this institution, which I learned in the old sports city of Castellana, it has been an honour to work for these colours," said Benitez.
"I would especially like to thank everyone at the new sports city and the Santiago Bernabeu who since my first steps through the door have supported me and made my work easier. Thanks to all for helping."
Real are third in La Liga, four points behind leaders Atletico Madrid and two behind champions Barcelona, who have a game in hand.
The team's biggest failure this season was a stinging 4-0 home defeat by arch-rivals Barca in November although they did qualify for the Champions League last-16 as group winners.
Real's next match is a La Liga game at home to Deportivo La Coruna on Saturday and they meet AS Roma in the first leg of their Champions League first knockout round tie on Feb. 17.
"I would like to wish good luck to Zinedine Zidane, my successor, and his staff," said Benitez who has also had stints in charge of Liverpool, Chelsea, Inter Milan, Valencia and Napoli.
"To all the players, coaches and staff ... I wish all the best and also good luck."
(Writing by Tony Jimenez; Editing by Justin Palmer)
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