* Exiled Hamas leader Meshaal says Israel failed

* Obama speaks to Middle East leaders

* Israel says it has pulled all troops out of Gaza Strip

* Gaza reconstruction bill estimated at nearly $2 billion

* Israeli army investigates use of white phosphorous

By Jeffrey Heller

JERUSALEM, Jan 21 (Reuters) - U.S. President Barack Obama pledged on Wednesday to pursue Middle East peace, telephoning Israeli and Palestinian leaders after Israel completed a troop withdrawal from the Hamas-ruled Gaza Strip.

In a call to Western-backed Palestinian President Mahmoud Abbas, Obama reiterated that he and his administration would work to achieve Middle East peace, a Palestinian official said.

Palestinian leaders later said they would only resume peace talks with Israel if the Jewish state commits to a comprehensive freezing of all settlement activity and undertakes to give up all occupied land captured in the 1967 Middle East war.

A statement from Olmert's office said the prime minister updated Obama on the situation in the Gaza Strip and added that he hoped efforts by Israel, Egypt, the U.S. and European countries to prevent weapons smuggling into Gaza would succeed.

The statement added that Olmert undertook that "Israel would invest in efforts to provide for the humanitarian needs of the Palestinian population in the Gaza Strip and would work to improve the economic situation in the West Bank".

In Washington, the White House said Obama had also spoken to Egyptian President Hosni Mubarak and Jordan's King Abdullah and that the U.S. president would actively engage in peace efforts.

"He used this opportunity on his first day in office to communicate his commitment to active engagement in pursuit of Arab-Israeli peace from the beginning of his term, and to express his hope for their continued cooperation and leadership," spokesman Robert Gibbs said in a statement.

Israel left the Gaza Strip devastated by its 22-day offensive. It completed its pullout earlier on Wednesday.

"We've redeployed on our side of the frontier and we will follow events closely," said Mark Regev, a spokesman for Olmert. "If Hamas breaks the ceasefire, we of course reserve the right to act to protect our people."

PRESSURE

Under international pressure to end the deadliest Israeli-Palestinian fighting in decades, Israel and Hamas declared separate ceasefires on Sunday, opening the way for more aid to be brought into the rubble-strewn enclave where thousands are homeless.

Reconstruction, if it can be launched in light of the West shunning Hamas as a "terrorist" group, may cost close to $2 billion, according to Palestinian and international estimates.

Diplomatic efforts led by Egypt were focusing on reaching a long-term Israel-Hamas truce deal, far short of an accord on Palestinian statehood sought by the United States and other international peace brokers.

Exiled Hamas leader Khaled Meshaal said in Damascus that Israel had failed to achieve its goals in Gaza.

"This battle has proved that force alone will not provide security for the Zionist entity (Israel)," Meshaal said.

He said that Arab countries seeking to help rebuild Gaza should donate money to the group's leader in the territory, Ismail Hanieyh, whom he described as the head of the legitimate Palestinian government, shunning Abbas's Fatah administration which holds sway in the occupied West Bank.

Israel's attacks in an offensive it began on Dec. 27 killed some 1,300 Palestinians. Gaza medical officials said the Palestinian dead included at least 700 civilians.

Israel said hundreds of militants died and that it dealt Hamas a strong blow that had boosted the Jewish state's power of deterrence and drawn international pledges to help prevent the Islamist group from replenishing its rocket arsenal.

Ten Israeli soldiers and three civilians, hit by cross-border rocket fire, were killed in the conflict.

Israel's Haaretz daily, reporting what it said were details of an army probe into its soldiers' use of white-phosphorous shells, said 200 were fired in the fighting, including 20 in a built-up area in the northern Gaza Strip.

Two Palestinian children were killed and 14 people suffered severe burns on Jan. 17 when Israeli shells landed in a U.N.-run school in the northern Beit Lahiya area, medical officials said.

TROOP WITHDRAWAL

Calling the troop withdrawal a "victory for Palestinian resistance", Hamas demanded a lifting of the blockade Israel tightened on the Gaza Strip after the Islamist group seized control of the territory from the Fatah movement in 2007.

Israel said at the start of the military campaign it never intended for its army, which quit the Gaza Strip in 2005 after 38 years of occupation, to remain there permanently.

Most Israeli forces pulled out before Obama was sworn in on Tuesday, in a move analysts saw as an attempt to avoid any early tensions with his administration.

Looking to reconstruction efforts, Israel has told the United Nations and aid groups they must apply for project-by-project approval and provide guarantees none of the work will benefit Hamas, Western and Palestinian officials said.

Israel, the officials said, is also preventing the Western-backed Palestinian Authority from transferring cash to the Gaza Strip to pay its workers and others hard-hit by war.

The restrictions threatened to undercut the ability of Abbas's government to reassert a presence in the enclave.

Hamas officials and an Israeli envoy planned to meet separately with Egyptian mediators in Cairo on Thursday to discuss ways to make the ceasefire "durable" and a reopening of border crossings, an official close to the talks said. (Additional reporting by Nidal al-Mughrabi and Doug Hamilton in Gaza, Adam Entous in Jerusalem and Ali Sawafta in Ramallah; Editing by Charles Dick)

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