The late-night host vowed to repair his relationship with his wife, Regina Lasko.
"Let me tell you folks, I got my work cut out for me," said Letterman, according to an early transcript of the program released by CBS.
Monday's show was the first Letterman taped since Thursday, when he disclosed that he had had sexual relationships with women who worked for him, and that he had been the victim of an alleged $2 million blackmail threat.
Letterman also apologized to his staff for the situation.
"Inadvertently, I just wasn't thinking ahead," Letterman said. "My thanks to the staff for, once again, putting up with something stupid I've gotten myself involved in."
Audience members who attended the taping Monday at the Ed Sullivan Theater said Letterman spoke frequently throughout the program about the events. He also joked about the situation with guests Steve Martin and Martin Short.
"He's not running away from it," said Brian Lakari, from Green Bay, Wis., who attended the taping with his wife, Kelly. "He brought it into the show several times."
Lakari said Letterman started his monologue with the subject, and even worked in an apology to "the previous governor of Alaska." He was referring to Sarah Palin, with whom Letterman had a flap earlier in the year over a joke about her daughters.
It is unclear how many women Letterman had sex with, and he has offered no specifics.
Robert Halderman, a producer for the true-crime show "48 Hours Mystery," pleaded not guilty last week to extortion charges.