Washington stood down from his role earlier in September, having led Texas since 2006 - including taking the ball club to two World Series match-ups.
At the time of his resignation, Washington said he needed to address an "off-the-field personal matter".
Washington's wife Gerry sat nearby at the press conference in which he revealed his reasons for quitting the Rangers' post.
"I was not true to my wife, after 42 years," Washington said.
"I broke her trust. I'm here today to own that mistake and apologise to her, and to those I disappointed, and those who have trusted in me, and I let them down.
"Today, I'm at a very low time in my life.
"I'm sorry for breaking the trust that I had with my wife and for disappointing my players, for disappointing my coaches, disappointing Major League Baseball and for disappointing the Texas Rangers."
Washington pleaded for privacy as he bid one final public farewell to Rangers fans, who have experienced a tough season that will see them likely finish bottom of the American League's West division.
"All I ask is for your forgiveness and your understanding," Washington said.
"I also ask that you respect our privacy as we go on with our lives. This matter is certainly personal and we are trying hard to put it behind us.
"I was born to be a baseball player. I'm a baseball lifer. The Rangers gave me a home, and I'm thankful for that.
"And I'm also thankful for the experience to have the opportunity to manage here in Texas."