"I come as an optimistic person and a realistic person - realistic in my understanding that it's vital for the world to fight terrorists, to confront those who would murder the innocent to achieve political objectives," Mr. Bush said.
Despite having promised throughout his term as president to press for peace in the Middle East, Mr. Bush's arrival in Jerusalem marks his first presidential visit to Israel.
Mr. Bush's staff says the timing of the visit shows his personal commitment to pushing the peace process forward, CBS News senior White House correspondent Bill Plante reports.
His first formal meeting was with Israeli President Shimon Peres, who said the peace conference Mr. Bush hosted late last year in Annapolis, Maryland, gave all parties in the difficult Mideast peace process one year to make progress. "Time is so precious," Peres said.
With Mr. Bush's days in office nearing their end, Peres said all sides must take this chance "extremely seriously."
Mr. Bush acknowledged the complexity of the task ahead for Israel.
"You know, its politics can be rough sometimes just like the politics of America can be rough," Mr. Bush said. "But nevertheless, we share a common vision of peace."
Mr. Bush is trying to build momentum for stalled Mideast peace talks and clear up confusion about whether the United States is serious about confronting Iran about its suspected nuclear ambitions.
Sources tell Plante that Israelis are most interested in finding out how the United States would react if Israel decided to attack Iran.
The president said both the United States and Israel have been targeted by terrorists, and he compared the battle against extremists to World War II when the U.S. and its allies fought Nazi Germany.
"I come with high hopes, and the role of the United States will be to foster a vision of peace. The role of the Israeli leadership and the Palestinian leadership is going to do the hard work necessary to define a vision," Mr. Bush said.
Peres underscored Mr. Bush's hopes - considered unrealistic by many in the Mideast - to bridge decades of differences in just one year and reach agreement for the establishment of a Palestinian state.
"The next 12 months will be a moment of truth," Peres told Mr. Bush at an airport arrival ceremony complete with red carpets and a military band. "It must not yield just words."
Mr. Bush offered an equally optimistic view of prospects for peace when he arrived earlier in Tel Aviv, saying "We see a new opportunity for peace here in the Holy Land and for freedom across the region."
Underscoring the importance of the visit, senior administration officials said for the very first time that they will blog every day about the trip on the White House Web site, Plante reports.
Unpopular at home, Mr. Bush was greeted here with smiles and warm handshakes. An honor guard and dozens of dignitaries, including the entire 26-member Israeli Cabinet, welcomed him.
"You are our strongest and most trusted ally in the battle against terrorism and fundamentalism and a staunch supporter of our quest for peace and stability," Prime Minister Ehud Olmert told the president.
Mr. Bush also stressed the deep U.S.-Israel ties.
"The alliance between our two nations helps guarantee Israel's security as a Jewish state," Mr. Bush said.
That remark lent support to Israel on one of the core issues in the conflict. The Palestinians oppose calling Israel a Jewish state, saying it rules out the right of Palestinian refugees to return to lost properties in Israel. They say the fate of the refugees is a matter for negotiations. Mr. Bush has referred to Israel as Jewish state in the past but the reference - here in the region - had special significance.
Pledging to stand with Israel against terrorists, Mr. Bush said, "We will do more than defend ourselves. We seek lasting peace."
Mr. Bush's challenge is to convince skeptical governments that, with just a year remaining in his presidency and Americans deep in the process of selecting his successor, he is willing to devote the time and effort necessary to bridge decades of differences in this troubled region.
On the eve of Mr. Bush's arrival, Olmert and Palestinian President Mahmoud Abbas pledged to have negotiators begin work immediately on the so-called final status issues. These include the final borders between Israel and a future Palestine, completing claims to the holy city of Jerusalem, the fate of millions of Palestinian refugees and Israeli security concerns.
Still, expectations of success are low, and no one is predicting big breakthroughs. There's been little headway since Mr. Bush hosted the splashy Annapolis conference in November, and launched the first major peace talks in seven years.
Just before Mr. Bush arrived, Palestinian militants in the Gaza Strip bombarded southern Israel with rocket and mortar fire, and Israeli troops killed a militant. Mr. Bush's three-day visit to Israel and the West Bank does not include stops areas near or in the Gaza Strip, which is controlled by Islamic Hamas militants who are not a party to negotiations.
The Palestinians are angry about Israeli plans to build new housing in east Jerusalem and the West Bank - areas captured by Israel in the 1967 Mideast war and claimed by the Palestinians for their future state.
In a boost to the Palestinians, U.S. Secretary of State Condoleezza Rice said in an interview published Wednesday that the U.S. considers the Har Homa neighborhood in east Jerusalem to be a settlement and opposes the new construction there.
"Har Homa is a settlement the United States has opposed from the very beginning," Rice told the Jerusalem Post, adding that the U.S. "doesn't make a distinction" between settlement activity in east Jerusalem and the West Bank. The comments were the Bush administration's strongest criticism to date of Israeli policies in east Jerusalem.
Israel, for its part, has demanded that Palestinian forces do more to rein in militants in the West Bank. Since Olmert and Abbas last met, two Israelis were killed in the West Bank, and Israeli security forces say members of Abbas' Fatah movement were responsible.