David Axelrod, another senior Obama adviser, said "obviously there are issues we need to discuss, such as reducing the high level of corruption" in Afghanistan, but he said "we are going to deal with the government that is there".
Axelrod pointed out that recent polling had indicated that Karzai was likely to win the runoff - scheduled for November 7 - anyway.
Obama and his national security advisers have been reviewing the US role in the eight-year war in Afghanistan, where 68,000 American troops will be fighting by the end of the year.
The war has turned increasingly bloody over the last several months, and the US president is considering a strategy shift to focus more on eliminating al-Qaeda and its allies in Pakistan with unmanned spy planes instead of sending many more troops to Afghanistan to target the Taliban.
The US and Nato commander in Afghanistan, General Stanley McChrystal, has asked for up to 80,000 additional troops and says he needs at least 40,000 more.
Al Jazeera's James Bays on the one-man race that is the Afghan presidential runoff vote
Axelrod repeated on Sunday that the decision on strategy was likely to come "within weeks".
Addressing his supporters on Sunday in the Afghan capital Kabul, Abdullah cited the government's refusal to make changes in the electoral commission for his decision to pull out.
He said a "transparent election is not possible" and that the Afghan government has been illegitimate since May.
"I'll not take part in the election," he said, adding that he had "not taken this decision easily".
In an interview with Al Jazeera shortly after the announcement, he said the decision was made after "a lot of consultations".
"It was the right decision, and I did it in the best interests of this country," Abdullah said.
"As far as the process is concerned, hopefully there is a way forward. I will not enter into the constitutional implications of this decision.
"But I'll be pursuing the agenda for change and reform in any capacity that I'll be. Perhaps, after a sigh of relief, I will enter into this role with my followers, with my supporters, with the movement that has supported me for the past five months."
Polls to continue
Shortly after Abdullah's announcement, Karzai's campaign team said the election would still go ahead.
"His withdrawal should not alter the process... The process should go on and the people of Afghanistan should be given the chance to vote," Wahid Omar, a Karzai spokesman, told Al Jazeera.
James Bays, Al Jazeera's correspondent in Kabul, said many Afghans had told him that an election with only one candidate would be "farcical".
He said one Afghan had told him that one of the great things that the West has always boasted about after the days of the Taliban, was bringing democracy to Afghanistan – but many Afghans were asking what sort of democracy results in a second-round election with only one candidate.
Haroun Mir, a political analyst in Kabul, said that the credibility of the election would depend on turnout.
"Everything will depend on the voter turnout," he told Al Jazeera.
"If the turnout is high, as in the first round, Karzai will be declared winner and he'll have legitimacy in the eyes of the Afghans and in the international community.
"But if the turnout is very low - below 25 per cent - Karzai will be declared legally as president, but will not enjoy strong legitimacy within the Afghan population or indeed the international community."
The first round of Afghanistan's elections on August 20 was so badly affected by ballot-box stuffing and distorted tallies that more than one million votes were thrown out.
The discovered fraud pushed Karzai below the required 50 per cent margin needed to win, forcing the country into a second round.
Following the election's first round in August, Abdullah had demanded that Karzai sack Azizullah Ludin, the head of the Independent Electoral Commission (IEC).
He also demanded the suspension of four ministers who campaigned for Karzai.