NO CEASE-FIRE CALL
While calling for an end to violence that has killed almost 120 Palestinians - around half of them civilians - Rice did not specifically demand a cease-fire.
The omission may anger some Arab allies who could see it as giving Israel a green light for more military action in Gaza.
Israeli troops pulled out of the Hamas-controlled Gaza Strip on Monday after international appeals, but a senior official said it was a "two-day interval" during Rice's visit and Olmert said Israel would not tolerate attacks.
Washington's role as an honest broker is under the spotlight in the current Gaza conflict because of its close ties to Israel. Rice came under heavy criticism during the 2006 Israeli-Lebanon war for not demanding a cease-fire.
Egypt has been looking at how to get one between Hamas and Israel but Rice declined to use the word cease-fire, which would involve directly negotiating with Hamas, a group Washington brands as terrorist and refuses to deal with.
"Call it what you will; we want the violence to stop," she said.
She strongly rejected parallels with the Lebanon conflict and said she would discuss with Egypt how to reduce the violence, as well as ways to secure Egypt's border with Gaza, briefly breached in January.
Rice said there was merit to the idea of giving the Palestinian Authority control at the sensitive border crossing at Rafah and having European monitors return. Hamas, however, has demanded a role at the border, largely closed since June.
"I will certainly be discussing how we make the Rafah situation sustainable and what that would mean for a role for the PA, for European monitors on that border," Rice said.
She also voiced concern about the loss of innocent life in Gaza and the humanitarian situation there, which she said would be brought up in all her discussions.
"I do think it will help if something can be done about the humanitarian situation in Gaza, including about regularizing in some fashion the Rafah crossing," she said.
Rice is set to move on to the West Bank to see Abbas and Palestinian Prime Minister Salam Fayyad later on Tuesday. She will then go to Jerusalem for talks with Israeli leaders, before leaving for Brussels on Wednesday for a meeting of NATO foreign ministers.
