On Monday, Israeli soldiers opened fire with automatic rifles across the Lebanese-Israeli border, according to Lebanese soldiers.
Israeli warplanes on Sunday bombed a suspected terrorist camp northwest of Damascus in retaliation for the suicide bombing the day before at a seaside restaurant in Haifa.
Mr. Bush decried the "needless murder" of 19 people in the suicide attack. Mr. Bush said that the Palestinian Authority must do more to fight terror and "must use whatever means is necessary. … All parties must assume responsibility."
The president commented after the new Palestinian prime minister, Ahmed Qureia, said he hopes to negotiate a quick truce with Israel, but won't use force against Palestinian militants under any circumstances — despite U.S. demands for a clampdown on armed groups.
The president was asked if he could work with a prime minister who would not use force against militants.
"We have not changed…the parties need to assume responsibility for their action in order for there to be peace," he replied.
During a White House news conference with Kenyan President Mwai Kibaki, Mr. Bush said that he had spoken with Sharon on Sunday.
"I made it very clear to the prime minister that…Israel's got a right to defend herself, that Israel must not feel constrained in terms of defense of the homeland," he said.
However, Mr. Bush added, "I said that it's very important all action should avoid escalation creating higher tensions."
Israel was on high alert Monday as Jews observed Yom Kippur, the Day of Atonement. The attack on Sunday was the first Israeli strike deep within Syria since the 1973 Yom Kippur War.
According to the Haaretz newspaper, Palestinian militants in refugee camps were on alert for other acts of retaliation by Israel.
Syrian villagers near the targeted site in Ein Saheb, 14 miles northwest of Damascus, said the camp had been used by Palestinian gunmen in the 1970s but was later abandoned.
The Arab League said the bombing "exposes the deteriorating situation in the region to uncontrollable consequences, which could drag the whole region into violent whirlpool."
It seemed unlikely Syria would retaliate. It has 380,000 active duty soldiers, but Israel holds a technological edge. Israel is more worried about Syria's growing missile program and its ability to launch chemical and poison weapons into Israel's cities.
Syria demanded that the U.N. Security Council condemn Israel's strike against a purported terrorist training camp near Damascus, but the United States said it would not support any resolution that does not also criticize attacks against Israel.
European Union foreign policy chief Javier Solana on Monday criticized the Israeli attack against an alleged terrorist training site in Syria, indicating that the strike was a violation of international law.
"I understand that the Israelis are suffering tremendous attacks from terrorists, suicide bombers. But as I said before, the reaction has to be within in the domain of international law," said Solana, who was in Rome after EU meetings. "Even if the Israeli has a right to defend himself from terrorism, it has to be done in the context and international rules and laws."
The Islamic Jihad claimed responsibility for the bombing but denied having training bases in Syria.
Earlier, White House spokesman Scott McClellan called the suicide bombing "despicable" but declined to weigh in on whether the suspected training camp was an appropriate target for retaliation.
He would not say whether the United States agreed with Israel's contention that the site was a training camp for terrorists, nor would he say whether the Mr. Bush administration would veto a U.N. resolution condemning Israel's airstrike.
"We've always stated that Israel has the right to defend herself," McClellan said, while cautioning the Israeli government to consider the "consequences" of its actions on the peace process.
Administration officials said Israel had not informed Washington in advance of its retaliatory strike nor indicated whether it intended any move against Palestinian leader Yasser Arafat to remove him from his West Bank headquarters.
Last month, the Israeli security Cabinet decided to remove Arafat "in principle," but has not launched any attempt to do so.
The U.S.-backed "road map" to Mideast peace has suffered a series of setbacks in recent months, with Israel building homes in new West Bank settlements in defiance of the plan and steady bombings by Palestinians.
"We always pointed out that there would be difficulties along the way," McClellan said.
Sunday, the State Department declared that Syria "must cease harboring terrorists and make a clean break from those responsible for planning and directing terrorist action from Syrian soil."
Syria says it has shut down the offices of Islamic Jihad and Hamas, two anti-Israel militant groups, but has not expelled its operatives. Meanwhile, the United States worries pro-Saddam Hussein fighters may be sneaking into Iraq across the Syrian border.