The former Manchester United and Everton manager's bid to resurrect his career in Spain ended in failure when Sociedad rescinded his contract following a defeat at Las Palmas which left the Basque club hovering just above the relegation places.
Moyes, 52, who was sacked by United in April 2014, was replaced by former Celta Vigo and Barcelona B coach Eusebio Sacristan with the team in similar dire straits to when the Scot took over a year ago.
In his first public comments since he was dismissed, Moyes said he regretted not being able to see out his contract, which ran until the end of the season.
He said he had "turned down a number of job opportunities in recent months" without specifying what they were.
"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," Moyes said in a statement published by the League Managers Association (LMA) on Tuesday.
"We succeeded in attaining that target and finished 12th in the table," he added.
"I feel that we made significant progress last season. Despite some positive performances from our team, results have not gone our way in recent weeks.
"However, I was confident we would finish in a strong position by the end of the season."
Moyes was considered a risky appointment by some in Spain because of his limited knowledge of La Liga and inability to speak the language, and he was criticised for not making more of an effort to learn Spanish.
He said he felt he had learned from working outside England.
"After spending 12 years in the Premier League as manager of Everton and Manchester United, I was keen to take the opportunity to work abroad and have learnt a great deal from the experience," he said.
(Reporting by Iain Rogers, editing by Rob Hodgetts)
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