The 62-year-old made the surprising on-air admission of having sex with female members of his staff last week, having been the alleged victim of a $2 million blackmail threat.
Speaking on his television show on Monday night, the Late Show presenter said his wife Regina Lasko had been 'horribly hurt' by his behaviour.
Letterman and Lasko were married in March this year after dating for more than 20 years and have a six-year-old son.
'When something happens like that, if you hurt a person and it's your responsibility, you try to fix it,' he told his studio audience in New York.
'And at that point, there's only two things that can happen: either you're going to make some progress and get it fixed, or you're going to fall short and perhaps not get it fixed, so let me tell you folks, I got my work cut out for me.'
Letterman claimed last Thursday an attempt had been made to blackmail him over his affairs with female employees, with a package received by the chatshow veteran's office claiming knowledge of the extramarital relationships.
Letterman contacted the Manhattan District Attorney's office and participated in a 'sting' operation, issuing a 'phoney' cheque for $2 million to Robert 'Joe' Halderman, a television producer, who was later arrested.
Mr Halderman's lawyer Gerald Shargel said on Monday his 48-year-old client was innocent of the alleged extortion and had not been portrayed fairly.
'David Letterman didn't give his [Halderman's] side of the story, David Letterman gave what he wanted the public to know,' Mr Shargel said on the Today show on NBC.
'He wanted to get out ahead of the story, and that's exactly what he did.'
If convicted, Mr Halderman, a producer of CBS show 48 Hours, faces up to 15 years in prison.
As well as apologising to his wife, Letterman also said sorry to his employees for involving them in his personal behaviour.
'Inadvertently, I just wasn't thinking ahead,' he added. 'My thanks to the staff for, once again, putting up with something stupid I've gotten myself involved in.'
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