In a statement made to the League Managers Association (LMA), the former Everton and Manchester United boss admitted his disappointment at having his La Real reign cut short despite helping the club finish 12th in La Liga during the 2014-15 campaign.
The Basque outfit only managed to pick up two wins this season leaving them in 16th place and Moyes was replaced by former Barcelona assistant coach Eusebio Sacristan.
In the statement, Moyes said: "I am disappointed to have left Real Sociedad. When I accepted the job as manager of La Real last November, my remit was to avoid relegation and retain the club's La Liga status.We succeeded in attaining that target.
"I made a firm promise to Real Sociedad to honour my contract and it is disappointing not to be able to fulfil this commitment especially when I have turned down a number of job opportunities in recent months.
"I would like to thank the club and the President for giving me the opportunity to manage in Spain, at a club with such a rich history of employing British Managers.
"I would also like to thank the players, staff and my assistant, Billy McKinlay, for their support and the wonderful people of San Sebastian."