Secretary of State Condoleezza Rice has also called for a durable cease-fire, and before departing on her peace-seeking mission to the Middle East and Rome, she said she would not engage in any “shuttle diplomacy” just for the sake of a cease-fire that would not last.

Mr. Bush has repeatedly talked of Israel’s right to defend itself against terrorism, even as he urged the country to show restraint and minimize civilian casualties. And the American State Department classifies Hezbollah as a terrorist group.

For his part, Mr. Maliki has condemned Israel for its campaign in Lebanon, but he has not condemned Hezbollah — and he declined an opportunity today to declare his position on the militant group, which has a political presence in the Lebanese government as well as a military arm that operates in the southern part of the country near Israel.

“Here, actually, we’re talking about the suffering of a people,” Mr. Maliki said when asked about Hezbollah. “We are not in the process of reviewing one issue or another or any government position.

“The important thing here is, what we are trying to do is to stop the killing and the destruction, and then we leave the room and the way for the international and diplomatic efforts and international organizations to play the role, to be there.” Mr. Bush and Mr. Maliki said they saw progress being made in Iraq despite the continuing violence there. They expressed confidence that the country will one day stand as an example of a peaceful democracy in a region that has known little peace or democracy.

“I have full confidence of victory, and we will be highly capable of defeating terrorism in Iraq,” Mr. Maliki said.

But Mr. Maliki’s refusal to condemn Hezbollah has created an awkward situation for the White House.

“His statements are troubling,” Senator Harry Reid of Nevada, the Democratic minority leader, said today. “They raise serious questions about whether Iraq, which is supposed to be our ally, can play a constructive role in resolving the current crisis and bringing stability to the Middle East.”