"The president needs to spell out a real and understandable plan for the unfinished work ahead" in Iraq, Senate Democratic Leader Harry Reid of Nevada said in remarks prepared for delivery Monday.
"Most of all, we need an exit strategy so that we know what victory is and how we can get there; so that we know what we need to do and so that we know when the job is done."
Another top Democrat, Sen. Edward M. Kennedy, D-Mass., went further, saying Mr. Bush "must look beyond the election" and start bringing troops home.
"The best way to demonstrate to the Iraqi people that we have no long-term designs on their country is for the administration to withdraw some troops now" and negotiate further withdrawals, Kennedy said in a statement.
The president did not mention any military withdrawal in his remarks Sunday, when he praised the bravery of Iraqis for turning out to vote "in great numbers and under great risk."
"The world is hearing the voice of freedom from the center of the Middle East," Mr. Bush said.
Secretary of State Condoleezza Rice also would not say whether U.S. forces will leave the country in great numbers after the vote. She said the United States will discuss the continued need for outside security forces with the newly elected Iraqi government.
Rice said the election went better than expected, but did not elaborate on U.S. predictions for turnout, violence or other measures.
Insurgents struck polling stations with a string of suicide bombings and mortar volleys, killing at least 44 people, including nine suicide bombers.
"Some Iraqis were killed while exercising their rights as citizens," Mr. Bush said. He also mourned the loss of American andkilled Sunday.
The president cautioned that the election will not end violence in Iraq.
"Terrorists and insurgents will continue to wage their war against democracy, and we will support the Iraqi people in their fight against them," Mr. Bush said. "We will continue training Iraqi security forces so this rising democracy can eventually take responsibility for its own security."
So far, more than 1,400 U.S. troops and many thousands of Iraqis have lost their lives. The United States is spending more than $1 billion a week in Iraq.
In Iraq, officials said turnout among the 14 million eligible voters appeared higher than the 57 percent they had predicted. Complete voting results are not expected for days.
Polls were largely deserted all day in many cities of the Sunni Triangle. In Baghdad's mainly Sunni Arab area of Azamiyah, the neighborhood's four polling centers did not open at all, residents said.
"It is hard to say that something is legitimate when whole portions of the country can't vote and doesn't vote," Sen. John Kerry, D-Mass., Mr. Bush's re-election challenger in November, said on NBC's "Meet The Press."