Photographs of US personnel sexually humiliating and physically abusing prisoners at the jail on the outskirts of Baghdad surfaced in April, triggering global condemnation and calls for Rumsfeld to quit.
"I submitted my resignation to President Bush twice during that period and told him that ... I felt that he ought to make the decision as to whether or not I stayed on. And he made that decision and said he did want me to stay on," Rumsfeld said.
He was speaking in an interview with CNN's Larry King Live programme on Thursday evening.
Rumsfeld did not state the specific dates when he offered his resignation.
On 5 May, a week after CBS broadcast the first Abu Ghraib pictures, White House aides said that Bush had complained to Rumsfeld that the secretary had failed to alert him fully to the details of the scandal.
But Bush the next day told reporters: "Secretary Rumsfeld has been the secretary during two wars and he's an important part of my cabinet and he'll stay in my cabinet."
"What was going on in the midnight shift in Abu Ghraib prison halfway across the world is something that clearly someone in Washington, DC, can't manage or deal with"
US Secretary of Defence
During testimony before Congress on 7 May, Rumsfeld said: "These events occurred on my watch. As secretary of defence, I am accountable for them. I take full responsibility."
At the same time, Rumsfeld and other Pentagon officials attributed the abuse to misconduct by low-ranking soldiers, several of whom have pleaded guilty or been convicted in military courts.
In his CNN interview, Rumsfeld said: "What was going on in the midnight shift in Abu Ghraib prison halfway across the world is something that clearly someone in Washington, DC, can't manage or deal with.
"I have no regrets," Rumsfeld said.
James Schlesinger (L) at a media
briefing on his investigation
"We've made a lot of corrections to make sure that those kinds of things (that) happened either don't happen again or are immediately found out and limited and contained," Rumsfeld added.
Last August an investigating panel named by Rumsfeld and headed by former Defence Secretary James Schlesinger said Pentagon leaders and the military command in Iraq contributed to an environment in which prisoners suffered sadistic abuse at Abu Ghraib.
The panel also found that Rumsfeld and top generals failed to exercise proper oversight over confusing detention policies at US prisons in Iraq and Afghanistan.
Abuse of detainees
Human rights activists argue that coercive interrogation techniques approved by Rumsfeld for use at the US naval base at Guantanamo Bay, Cuba, where foreign terrorist suspects are held, helped set the stage for detainee abuse by US forces.
Rumsfeld, a combative 72-year-old former Princeton University wrestler and navy fighter pilot, has established himself as the most powerful Pentagon chief since Vietnam war-era Defence Secretary Robert McNamara.
US soliders put Rumsfeld on the
spot over their equipment
While he has many admirers among Bush supporters, Rumsfeld has amassed numerous enemies as well and many Democrats have called for him to quit.
Critics pounced in December after Rumsfeld told a soldier who asked about a lack of armored US vehicles in Iraq that "you go to war with the army you have". He also admitted his letters to families of troops killed in Iraq had been signed by a machine.