On the ground, insurgents renewed their attacks after a period of relative calm following the polls that surprised the world with their high turnout despite a string of bombings and mortar attacks that killed at least 36 people.
Iraq's President Ghazi Al-Yawar said on Tuesday it would be "complete nonsense" to ask US and other foreign troops to leave the country now.
As the massive security election clampdown was eased, a Shiite political leader escaped an assassination attempt in the holy city of Najaf, at least eight people were killed in a string of attacks around the so-called Sunni triangle and an oil pipeline was sabotaged.
Iraqi and US officials have urged the country's various communities to unite after the polls, the first since Saddam Hussein was toppled in a US-led war in 2003 and the first free election in more than half a century.
But the Committee of Muslim Scholars, the premier organisation of Sunni cleris across Iraq, poured cold water on hopes for an end to the community's hard-line stance against the US-imposed experiment of democracy in Iraq. The association, which called for a Sunni boycott of the election, said the next government would lack the authority to write a new constitution laying out the framework for a post-Saddam future.
"These elections lack legitimacy because a huge portion of the population boycotted and this tells us the national assembly and the coming government will not have the legitimacy required for writing the constitution, or concluding security and trade agreements," the committee said.
It stopped short of slamming the door and offered a tentative olive branch to the next government in acknowledgement of the millions of Iraqis who braved Sunday's violence, saying it would have "limited authority." As Iraq waited for the final vote tally, the election commission said it was investigating remedies to the fact tens of thousands of people who were unable to vote due to a short of ballot papers.
Yawar told a press conference on Tuesday that "tens of thousands were not able to vote for the lack of ballots", with the problems mainly in Mosul, Basra, Baghdad and Najaf.
Abdul Hussein al-Hindawi, chairman of the electoral commission, also acknowledged problems in Sunni areas where polling stations had not opened or not enough ballots had been distributed in Nineveh, Salaheddin and Tamim provinces.
In an effort to reach out to the disenchanted Sunni population, Yawar, a tribal Sheikh from Mosul, insisted there was consensus that his post would be retained by his fellow Sunnis in the next cabinet.
Yawar said he believed the ethnic shareout would remain the same for the post-election government that must oversee the drawing up of a new constitution, with the Kurds also being given the post of national assembly speaker.
In Washington, Rice said the goal was to get Iraq up and running quickly and gradually scale back the American presence.
"I don't think we want to talk in terms of exit strategies," she said. "I think we should ask ourselves what can we do now to - as quickly as possible - make the Iraqis self-sufficient but also give them the support that they need."
She hailed the Iraq election as opening the door to greater international involvement in the country.