Monk was devastated to be axed by Swansea on Wednesday after nearly two years in charge of the Welsh team.
The 36-year-old, who last season guided Swansea to a best-ever Premier League finish, conceded he had run out of time after a run of only one win in 11 games left them precariously perched a point above the relegation zone.
"It's with great sadness that I am leaving this special club," Monk said in a statement released by the League Managers Association.
"I believe that whenever you join a football club, the supporters deserve your blood, sweat and tears. I can say for sure I gave all those things and more.
"The transition from player straight to manager in the biggest league in the world is never an easy challenge, but it is a challenge that I have relished and immersed myself in.
"I have always tried to improve the principles and values that we have built at this club on and off the pitch on a daily basis."
Monk, who succeeded Michael Laudrup in February 2014, is convinced he has left a strong enough squad to ensure Swansea don't lose their top-flight status.
"I have no doubt whatsoever that these players will start to show their true quality again very soon, and the club will be looking upwards once more," Monk said.
"I feel truly proud of all we've done during my 22 months in charge."
Alan Curtis is set to take charge of Swansea on a caretaker basis against Manchester City on Saturday as club chairman Huw Jenkins continues his search for a new manager.