"I understand it's going to be difficult to achieve peace. But I believe it can happen," Mr. Bush told reporters after meeting at his ranch with Japanese Prime Minister Junichiro Koizumi.
The president said his administration's pledge to fully and seriously address Israel's security concerns under a new peace plan gave a crucial nudge toward Israel's formal acceptance of a formula for accord with the Palestinians.
"Prime Minister Sharon accepted the road map, and that's progress," Mr. Bush said.
Sharon said Friday he will present the plan to the Cabinet for approval, probably early next week, reports CBS News Correspondent Robert Berger, even though it's clear he doesn't like it. If implemented, the road map's call for a freeze on Jewish settlement activity could cause the Sharon government to collapse.
The president said he is "exploring the opportunities" for meeting with Sharon and Palestinian Prime Minister Mahmoud Abbas in the near future. Mr. Bush is considering a three-way meeting with Sharon and Abbas in Egypt, officials involved in the planning said.
"If a meeting advances progress toward two states living side by side in peace, I will strongly consider such a meeting," Mr. Bush said. "I'm committed to working toward peace in the Middle East."
Before Mr. Bush spoke, Secretary of State Colin Powell and Condoleezza Rice, the national security adviser, released a statement stressing that there is room to hear contrasting views on the peace plan. But at a news conference in Paris, Powell said: "With respect to the road map, we're not planning on making any changes to the road map."
"We have told the Israeli government that we would take their comments and address them seriously and fully as we went forward in the implementation of the road map," Powell, who was meeting in Paris with French Foreign Minister Dominique de Villepin, told reporters.
"This does not require us to change the road map. It is a good document that leads to the president's vision of two states living in peace side by side," Powell said.
Koizumi lauded Mr. Bush's effort to spur renewed peace efforts in the Middle East, and said "the world should cooperate together" to make the region more peaceful and stable. He said he will visit Egypt and Saudi Arabia in coming days to engage Arab nations on Iraqi reconstruction.
Sharon said Friday he is "prepared to accept" the U.S.-backed peace plan and will present it to the Cabinet for approval.
Palestinian Information Minister Nabil Amr told The Associated Press: "We are ready to start the implementation process immediately. We have got American confirmation that there will not be any changes in the road map."
The chain of events represented progress for Mr. Bush as he considers a new stepped-up personal investment in Mideast peacemaking.
Powell and Rice said in their statement that the president asked the Israeli government to respond to the road map "with contributions to this document to advance true peace." In that response, the pair said, the Israelis explained their "significant concerns."
Before the Bush White House announced the understanding with Sharon, a pipe bomb went off Friday near an armored bus carrying Israelis in the Gaza Strip in an attack claimed by the Islamic militant group Hamas.
A wave of such attacks caused Sharon to postpone his trip to the White House this week.
One of Sharon's main worries is that the road map, which requires the dismantling of recently established Israeli outposts, might be construed as requiring the dismantling of all Jewish settlements on the West Bank. Sharon told the Israeli people last week that abandoning settlements was not on the horizon.
Palestinian officials said Friday they would not accept any changes to the peace proposal and that they'd been assured by the Bush administration there would be none.
"We are ready to implement the road map as one package ... and without any changes," said Amr.
The road map is an attempt to end 32 months of fighting, freeze construction of Jewish homes on the West Bank and establish a Palestinian state by 2005 on land Israel has held for 36 years.
On the Palestinian side, the administration has been seeking an end to attacks on Israel. Mr. Bush has appealed to Abbas to clamp down on terror attacks while also reassuring him the administration still intends to help create a Palestinian state in 2005.