The former England boss was shown the door by the Foxes on Monday after a year in charge. Leicester have won just five games this season, but they are still only two points adrift of the play-offs. However, the club's Thailand-based owners lost patience after funding a £10million spending spree over the summer.
In a statement on the League Managers' Association website, Eriksson said: "I am disappointed to be leaving Leicester City Football Club. It was always my aim to lead the club into the Barclays Premier League and I am very sorry that I won't be able to do that. I have always thought, and still believe now, that the club will be promoted this season."
The statement continued: "The current squad is capable and ready to achieve promotion, but football is a results industry and unfortunately we couldn't quite get things right enough of the time.
"Being the manager of Leicester City was an absolute pleasure and I wish everyone connected with the club the best of luck. Leicester City is a great club and I am certain that with the current owners in charge the club will go from strength to strength.
"To the players, old, new, everyone in the squad, I would like to say thank you. You are a talented group and I am hopeful that you will achieve success very soon. To the staff at the club, both at the stadium and the training ground - you treated me like a king and I shall miss working with you all.
"Last but not least, to the fans, I want to say a huge thank you for your support. You are fantastic people and I will never forget the way you welcomed me into your club. I am just sorry that I won't be on the bench the day you reach the Premier League."
Former Foxes manager Martin O'Neill is an early front-runner, with the bookmakers at least, to return to the club after 11 years. In the meantime academy director Jon Rudkin and coach Mike Stowell have been placed in temporary charge ahead of Saturday's trip to West Ham.
Leicester and West Ham were installed as the pre-season favourites for promotion and the Hammers, who have also overhauled their squad under Sam Allardyce, moved up to second following Monday night's win at Brighton.
Allardyce said: "It's a results-driven industry today, and results have to come quickly. The problem with that is you go out and bring in players - I brought in 14 - and it's such a difficult job to produce results until you get to know the players and they get to know each other and you build up team spirit. The results take time."