Hamas threatened unprecedented revenge, with spokesman Abdel Aziz Rantisi saying Israel had "opened the gates of hell" with the attack on Hamas founder, Sheik Ahmed Yassin, who escaped with a minor injury.
Israeli officials say the bomb was smaller than usual to avoid collateral damage, reports CBS News Correspondent Robert Berger.
Israel declared a high security alert and imposed a blanket closure on the West Bank and Gaza Strip. Security officials said the Shin Bet security service tightened protection of Israeli leaders.
Sharon told the Yediot Ahronot newspaper that Israel's campaign against militants would continue.
"They are marked for death," Sharon was quoted as saying, referring to Hamas leaders. "We won't give them a moment's rest. We will continue to hunt them because they have only one objective: the destruction of Israel."
Arafat intends to ask parliament speaker Ahmed Qureia to become the new Palestinian prime minister, a Palestinian political source said Sunday.
Arafat made his choice of Qureia clear at a meeting with the leadership of his Fatah party in Ramallah, the Palestinian seat of government, during a day of political back-and-forth set off by Saturday's resignation of Abbas.
It was not clear when Qureia would be formally tapped for the position, the source said, speaking on condition of anonymity.
Abbas resigned after Arafat refused to grant him more power over the Palestinian security services.
Arafat has not yet accepted the resignation in writing, as required by law, but told Palestinian lawmakers he considered Abbas' Cabinet a caretaker government, implying recognition of the resignation. Earlier on Sunday, however, Arafat aide Nabil Abu Rdeneh said the leader hoped to persuade Abbas — also known as Abu Mazen — to remain and form a new government.
"Abu Mazen remains Arafat's first choice. But if he insists on maintaining his resignation, there will be a new appointment, and that will be discussed now and tomorrow," Abu Rdeneh said, emerging from a closed-door meeting where Arafat and leaders of his Fatah movement were discussing options. "Within 48 hours, we will reach a conclusion."
Abbas insisted his decision was final on Saturday and told a confidant he felt abandoned by all sides and was deeply hurt. But a source close to Abbas said Sunday he might serve again if he reached a firm agreement with Arafat on his powers and on the composition of the government beforehand. The source, who spoke on condition of anonymity, said this would be difficult.
At Sunday's meeting, many Fatah officials said they hoped to convince Abbas to serve again, said Hussein Sheikh, a West Bank Fatah leader.
Abbas' resignation left Israel and the United States without a negotiating partner, at least temporarily, and dealt a severe blow to the already troubled U.S.-backed "road map" peace plan. The two nations refuse to deal with Arafat, saying he is tainted by terror and an obstacle to peace.
If the resignation becomes final, Arafat has three weeks to name a successor. Among possible candidates is parliament speaker Ahmed Qureia, who is seen as a moderate and has credibility with Israel because he was an architect of the 1990s peace accords.
In coming days, the Israeli Cabinet also will reconsider possible action against Arafat, including sending him into exile, said Foreign Minister Silvan Shalom.
"As long as Arafat is in the region, he won't let any other leader develop," Shalom told Israel Army Radio.
Government spokesman Dore Gold said all blame fell to Arafat "who refused to let Mahmoud Abbas rule as prime minister, who refused to let him dismantle the infrastructure of terrorism."
The United States has blocked Arafat's expulsion in the past, and Sharon's security advisers have warned that Arafat could do more harm to Israel abroad than by remaining trapped at his West Bank headquarters in the town of Ramallah.
Israel's airstrike against the Hamas leadership on Saturday came just hours after Abbas announced his resignation.
A top Israeli security official said Abbas' departure released Israel from the last restraints in its war on the militants. With Abbas still in office, concern about harming his standing with airstrikes had always been a consideration, the official told The Associated Press on condition of anonymity.
Security officials said Sharon and his military chiefs learned early Saturday that Hamas leaders, including Yassin, his top aide Ismail Hanieh and chief bombmakers Mohammed Deif and Adnan al-Ghoul would meet at the apartment of a Hamas activist, Dr. Marwan Abu Ras, later in the day.
Defense Minister Shaul Mofaz, army chief Lt. Gen. Moshe Yaalon, Shin Bet director Avi Dichter and Air Force chief Maj. Gen. Dan Halutz rushed to the army's Tel Aviv headquarters to supervise the operation, the officials said. Sharon kept in touch by phone, though he was reportedly busy at the time with preparations for the birthday party of his grandson Rotem.
By mid-afternoon Saturday, some 10 top Hamas members were assembled at the home of Abu Ras, a university lecturer, the Israeli officials said. With Sharon giving the final go-ahead, an F-16 fighter plane dropped a laser-guided bomb on the home.
The top floor was still under construction, and Yassin and the others apparently were on the ground floor. Sixteen people in the house were hurt, including Yassin.
Yassin denied he was chairing a Hamas leadership meeting at the time of the strike.
Arafat called Yassin on Saturday evening and congratulated him on his survival, Hanieh said.
After being whisked away from Abu Ras' house, Yassin surfaced later Saturday at a Gaza City mosque and threatened revenge. Sharon "has to understand that he will pay the price for all his crimes, and the Israeli people will pay a high price as well," Yassin told thousands of supporters.
However, the Hamas military wing, Izzedine al-Qassam, clarified Sunday that it was not threatening Sharon personally. "Each Zionist who occupies our land is a target for us, but we did not select a specific target and we leave this to the judgment of our fighters and their ability to reach targets," the statement said.
About 3,000 students from the Islamic University marched in Gaza City on Sunday in support of Yassin, waving pictures of the bearded, quadriplegic leader.