The last few days have seen an upsurge in violence
Mr Bush was speaking after Israel said it would continue attacks on Palestinian targets in the West Bank and Gaza Strip, in retaliation for the weekend suicide bombings in Israel.
Just hours later, there was an explosion in central Jerusalem - apparently caused by a suicide bomber who blew himself up.
BBC correspondent Paul Adams, who is staying in a hotel opposite the blast, said he could see what appeared to be the remains of a body lying in the road.
Witnesses at the scene said at least three other people were lightly injured in the attack.
Mr Bush said: "It's incumbent upon other friends and allies of ours around the world to help bring those terrorists to justice if we want peace in the Middle East.
"I think the Israeli people want to have peace, but we learned in such a vivid way that there are elements in the Middle East that hate the thought of peace and will be willing to use terror to derail any type of peace process."
Mr Bush had earlier frozen the assets of a US-based charity and two other groups accused of financing Hamas, the Islamic militant group that claimed responsibility for the suicide bombings.
Tuesday's Israeli air strikes on Palestinian targets in the West Bank and Gaza claimed at least two lives and injured more than 150, including dozens of schoolchildren.
Mr Arafat, whose movements have been limited since Israel destroyed his helicopters and airport in Gaza, escaped injury when missiles hit his Ramallah headquarters in the West Bank.
The air strikes stopped on Tuesday afternoon, possibly due to stormy weather, and it was not clear when they would resume.
But a spokesman for the Israeli army, Brigadier General Ron Kitri, said military action would continue until the Palestinian leadership took effective action to stop attacks against Israeli civilians.
"I think [the attacks] will end only when we are convinced that
the Palestinian Authority is doing its duty and fighting terrorism," General Kitri told Israel radio.
In his first public statement after the attacks, Mr Arafat accused Mr Sharon of trying to torpedo his crackdown on militants.
"They [the Israelis] don't want me to succeed and for this he [Mr Sharon] is escalating his military activities against our people, against our towns, against our cities, against our establishments," Mr Arafat said.
"He doesn't want a peace process to start," he said.
Cabinet rift
The latest attacks followed a statement by the Israeli cabinet declaring the Palestinian Authority an "entity that supports terror".
The declaration has caused a rift in Mr Sharon's unity cabinet, with Foreign Minister Shimon Peres accusing elements within the government of attempting to bring about the downfall of the Palestinian Authority.
Mr Peres, the Labor leader, said Arafat "should be given a chance," suggesting that if he were, the Palestinian leader would be able to take a stand against militant groups.
The Labor Party is scheduled to convene on Wednesday to decide whether or not to pull out of Mr Sharon's wide national unity coalition in light of the military escalation.
Palestinian Information Minister Yasser Abed Rabbo denied Mr Sharon's charges of terrorism, saying that Israel's occupation of the West Bank and Gaza Strip was "the source of terrorism", and that Israel's actions were making the prospects of peace more remote.