The former basketball star issued the apology through publicist Jules Feiler in an email message to The Associated Press, a day after he sang "Happy Birthday" to North Korean leader Kim Jong Un at the start of the friendly game.
Rodman has been slammed for not using his influence with Kim to help free Kenneth Bae, the missionary in poor health who is being confined in North Korea for "anti-state" crimes. In an interview with CNN on Tuesday, Rodman implied Bae was at fault.
"I want to apologize," Rodman said Thursday. "I take full responsibility for my actions. It had been a very stressful day. Some of my teammates were leaving because of pressure from their families and business associates. My dreams of basketball diplomacy was quickly falling apart. I had been drinking. It's not an excuse but by the time the interview happened I was upset. I was overwhelmed. It's not an excuse, it's just the truth."
Rodman said he wanted to apologize first to Bae's family. "I'm very sorry. At this point I should know better than to make political statements. I'm truly sorry."
In the interview, Rodman was asked whether he would raise the issue of Bae during his visit.
"Kenneth Bae did one thing," Rodman replied. "If you understand what Kenneth Bae did - do you understand what he did in this country?"
Asked to explain, Rodman declined to respond.
Bae, a Korean-American Christian missionary and tour operator based in China, has been detained for more than a year. North Korea sees missionary work as a threat to its authoritarian government.
Bae's sister, Terri Chung, said his family couldn't believe what Rodman had said.
"Here's somebody who is in a position to do some good for Kenneth and refuses to do so," Chung told KOMO Radio in Seattle on Wednesday. "And then after the fact, instead, he decides to hurl these unqualified accusations against Kenneth. It's clear he has no idea what he's talking about. I'm not sure who he's talking to, where he's getting his information, but he's certainly no authority on Kenneth Bae."
The U.S. State Department distanced itself from Rodman and said it did not want to "dignify" his activities or comments in Pyongyang by commenting on them. But spokeswoman Jen Psaki said the department was open to speaking with Rodman on his return.
"We have not reached out to him. We've said before, if he wants to reach out to us, we're happy to hear from him and what he has to say," she told reporters.
Rodman dedicated the game to his "best friend" Kim, who along with his wife and other senior officials and their wives watched from a special seating area. The capacity crowd of about 14,000 at the Pyongyang Indoor Stadium clapped loudly as Rodman sang a verse from the birthday song and then bowed deeply to Kim, seated above him in the stands.
Rodman said he was honored to be able to play the game in the North Korean capital and called the event "historic." Some members of the U.S. Congress, the NBA and human rights groups, however, say he has become a public relations tool for North Korea's government.
The government's poor human rights record and its threats to use nuclear weapons against rival South Korea and the United States have kept it a pariah state. Kim shocked the world in December by having his uncle, once considered his mentor, executed after being accused of a litany of crimes including corruption, womanizing, drug abuse and attempting to seize power.
Rodman has refused to address those concerns while continuing to forge a relationship with Kim.
Rodman is the highest-profile American to meet Kim, who inherited power after the death of his father in late 2011. Rodman has said he is not a statesman and instead is seeking only to build cultural connections with the North through basketball that may help improve relations between Pyongyang and Washington.
Along with Rodman, the former NBA players included ex-All Stars Kenny Anderson, Cliff Robinson and Vin Baker. Also on the roster were Craig Hodges, Doug Christie, Charles D. Smith and four streetballers.
Associated Press writer Doug Esser in Seattle, Washington, contributed to this report.