Mr. Bush did say he would send CIA Director George Tenet back to the region to help build a Palestinian security force to fight terrorism.
The president, ending an Oval Office meeting that lasted more than an hour, renewed his call for a separate state for the Palestinian people.
Sharon replied, "I think it's premature to discuss" that issue until Palestinian leader Yasser Arafat reforms his government.
Just as the talks started in Washington, a suicide attack shattered a pool hall in the Israeli city of Rishon Letzion, killing more than 15 people and wounding at least 60, police said.
After the attack, Sharon announced he was cutting short his visit to Washington, canceling a scheduled meeting with congressional leaders.
CBS News Chief White House Correspondent John Roberts reports Mr. Bush and Sharon did not discuss the latest attack during their meeting. But later Tuesday, an aide said Mr. Bush was "disgusted" by the bombing, calling it a "wanton taking of innocent life," White House national security adviser Condoleezza Rice said.
The attack overshadowed a glimmer of good news in the region: Israel and the Palestinians agreed to terms for ending a 36-day standoff at the Church of the Nativity. A deal there could prompt Israel to withdraw troops from Bethlehem, satisfying Mr. Bush's monthlong demand to end incursions into Palestinian cities.
Sharon cited that advance as a cause for hope. "I feel there is a chance now to start a move forward," he said.
Mr. Bush, too, expressed optimism that Israel, the Palestinian Authority and moderate Arab leaders will join in efforts to curb terrorist attacks in Israel while opening negotiations toward a Palestinian state. He said Arab leaders, in particular, appear to be stepping up to a responsibility to press Arafat to reform his regime — making it more democratic and less corrupt.
"The world is rallying toward these reforms," Mr. Bush said.
The two leaders entered Tuesday's talks with sharply differing agendas. Sharon wants to cut Arafat out of the Mideast negotiating process. But the president maintains, for all his faults, that Arafat still represents the Palestinian people.
Mr. Bush also believes Saudi Arabia is a key peace partner, while Sharon's government has accused the Saudis of supporting terrorism.
Rather than emphasizing the disagreements, the president focused on Tenet's bid to build a security force that could crack down on terrorism.
Mr. Bush said the security force must be a united Palestinian unit that "is held accountable."
"I think it is very important that there be a unified security force, but at the same time we need to work for other institutions, a constitution for example, development of a state that can help bring security and hope to the Palestinian people and the Israelis," he said.
The reforms must begin as soon as possible, he said, and the United States will ask other Arab leaders to help.
"It's very important for us to seize this moment and lead," Mr. Bush said.
Sharon said he will be a dedicated peace partner. "We are committed to taking every effort and every step to reach peace," he said.
U.S. officials said Mr. Bush also was pushing Israel to ease economic restrictions on the Palestinians.
The Oval Office talks, the fifth meeting between the two leaders, came amid a flurry of diplomatic activity.
Secretary of State Colin Powell sought to seal the Bethlehem agreement in telephone calls to Italian President Silvio Berlusconi, whose country is the most likely candidate to receive 13 men due to be deported after leaving the church.
Mr. Bush worked the telephones, too, calling Egyptian President Hosni Mubarak and Crown Prince Abdullah of Saudi Arabia. The president, who meets Wednesday with Jordan's King Abdullah II, is urging moderate Arab leaders to press Arafat for action against terrorism.
All three parties are waging public relations battles while balking at Mr. Bush's demands. Arab leaders are urging him to be tougher on Sharon and more sympathetic to Palestinian needs. Israel wants Bush to treat Arafat like a terrorist.
The White House sought to strike a balance, with carrots and sticks for both Israel and the Palestinians.
Mr. Bush himself mentioned "Palestine," the name of the hoped-for state.
The president was asked if he would demand that Sharon deal with Arafat.
"I'm never going to tell my friend the prime minister what to do," he said.
A senior administration official, speaking on condition of anonymity, said Mr. Bush was pressing Sharon to ease economic restrictions on Palestinians. Among his ideas, the president wants Israel to release sales taxes impounded since Jan 2001. Under the 1993 Oslo agreement, the taxes are supposed to be turned over to the Palestinian Authority.
Sharon has previously rejected U.S. calls to release the taxes, saying some of the money would go to terrorists.
The United States has pledged more than $100 million in aid to the Palestinians. Mr. Bush has urged Arab leaders to help ensure that international aid gets to the Palestinians without getting snagged in what he considers to be a corrupt Arafat regime, U.S. officials said.
While Mr. Bush has expressed disappointment in Arafat, the White House says Israel must deal with him.
At every stop on his visit, Sharon offered evidence contending that Arafat and Saudi Arabia help terrorists.
White House spokesman Ari Fleischer gently turned both indictments aside. He said Arafat has not earned Mr. Bush's trust but stopped short of calling him a terrorist, and he said the administration accepted Saudi Arabia's assurances that the country does not support terrorists.
Saudi Crown Prince Abudullah's foreign policy adviser, Adel Al-Jubeir, accused Sharon of trying to derail peacemaking with the accusations. "Our support is not targeted to suicide bombers," he said.
Privately, White House officials said Mr. Bush believes Arafat has had a hand in terrorism and noted that Saudi Arabia is a hotbed for militants. But they said the president has set those concerns aside — at least publicly — because Arafat is the Palestinian leader and Saudi Arabia has broken new ground by outlining a plan under which Israel would have normalized relations with the Arab world.