President Barack Obama, speaking by phone Sunday with Israeli Prime Minister Benjamin Netanyahu, stressed the need for an immediate, unconditional humanitarian ceasefire in Gaza, the White House said.
Urging a permanent end to hostilities on the basis of the 2012 ceasefire agreement, Obama added that "ultimately, any lasting solution to the Israeli-Palestinian conflict must ensure the disarmament of terrorist groups and the demilitarization of Gaza."
Fighting subsided in Gaza earlier Sunday after Hamas Islamist militants said they backed a 24-hour humanitarian truce.
Hamas said it had endorsed a call by the United Nations for a pause in the fighting in light of the upcoming Muslim holiday of Eid al-Fitr, expected to start in the next couple of days.
Some firing of rockets continued after the time that Hamas had announced it would put its guns aside and Netanyahu questioned the validity of the truce.
Israeli artillery guns also fired barrages into the Gaza Strip, Israeli media reported, although the object of the fire was initially unclear.
"Hamas doesn't even accept its own ceasefire, it's continuing to fire at us as we speak," Netanyahu said in an interview with CNN, adding that Israel would "take whatever action is necessary to protect our people".
Nonetheless, Gaza Strip residents and Reuters witnesses said Israeli shelling and Hamas missile launches had slowly subsided through the afternoon, suggesting a de facto truce might be taking shape as international efforts to broker a permanent ceasefire appeared to flounder.
However, Israel's military has said it will need more time to destroy a warren of tunnels that criss-cross the Gaza border that it says is one of its main objectives.
Egypt had also destroyed 13 tunnels which crossed into its territory, an Egyptian general said on his Facebook page. It was "a continuation of the efforts by the armed forces in protecting the borders of the state from smugglers and terrorists," Brigadier General Mohamed Samir Abdulaziz Ghoneim said.
Israel and the Hamas Islamists who control Gaza had agreed to a 12-hour ceasefire on Saturday to allow Palestinians to stock up on supplies and retrieve bodies from under the rubble.
Netanyahu's cabinet voted to extend the truce until midnight on Sunday at the request of the United Nations, but called it off when Hamas launched rockets into Israel during the morning.
Palestinian medics said at least 10 people had died in the wave of subsequent strikes that swept Gaza, including a Christian woman, Jalila Faraj Ayyad, whose house in Gaza City was struck by an Israeli bomb.
Some 1,031 Palestinians, mainly civilians and including many children, have been killed in the 20-day conflict. A Gaza health ministry official issued revised figures of dead, saying that 30 fewer people than thought had died in the conflict.
Israel says 43 of its soldiers have died, along with three civilians killed by rocket and mortar fire out of the Mediterranean enclave.
Israel launched its Gaza offensive on July 8, saying its aim was to halt rocket attacks by Hamas and its allies.
After aerial and naval bombardment failed to quell the outgunned guerrillas, Israel poured ground forces into the Gaza Strip 10 days later, looking to knock out Hamas's rocket stores and destroy the vast network of tunnels.
The army says its drive to find and eliminate tunnels would continue through any temporary truce.
Diplomatic efforts led by U.S. Secretary of State John Kerry to end the 20-day conflict have shown little sign of progress. Israel and Hamas have set conditions that appear irreconcilable.
Hamas wants an end to the Israeli-Egyptian economic blockade of Gaza before agreeing to halt hostilities. Israel has signaled it could make concessions toward that end, but only if Gaza's militant groups are stripped of their weapons.
"Hamas must be permanently stripped of its missiles and tunnels in a supervised manner," Economy Minister Naftali Bennett said, "In return we will agree to a host of economic alleviations," the security cabinet member said on Facebook.
Kerry flew back to Washington overnight after spending most of the week in Egypt trying to bridge the divide, putting forward some written proposals to Israel on Friday.
Speaking off the record, cabinet ministers described his plan as "a disaster", saying it met all Hamas demands, such as lifting the Israeli-Egyptian blockade completely and ignored Israeli terms, such as stripping Hamas of its rockets.
There was no immediate comment from U.S. officials.
The obvious rancor added yet another difficult chapter to the already strained relations between Netanyahu and Kerry, whose energetic drive to broker a definitive peace deal between Israel and the Palestinians ended in acrimony in April.
The main U.N. agency in Gaza, UNRWA, said 167,269 displaced Palestinians have taken shelter in its schools and buildings, following repeated calls by Israel for civilians to evacuate whole neighborhoods ahead of military operations.
But in southern Gaza, residents of villages near the town of Khan Younis attacked the offices of the International Committee of the Red Cross, torching furniture and causing damage, saying the organization had not done enough to help them.
During the lull in fighting inside Gaza on Saturday, residents flooded into the streets to discover scenes of massive destruction in some areas, including Beit Hanoun in the north and Shejaia in the east.
An Israeli official said the army hoped the widespread desolation would persuade Gazans to put pressure on Hamas to stop the fighting for fear of yet more devastation.
The Israeli military says its forces have uncovered more than 30 tunnels in Gaza, with some of the burrows reaching into Israeli territory and designed to launch surprise attacks on Jewish communities along the border.
The military said on Sunday it found a tunnel that led directly into the dining room of an Israeli kibbutz.
Other underground passages, the military says, serve as weapons caches and Hamas bunkers. One official said troops had found it easier to operate during the truce as the immediate threat to their safety was diminished.
The Gaza turmoil has stoked tensions amongst Palestinians in mainly Arab East Jerusalem and the occupied West Bank.
Medics said eight Palestinians were killed on Friday in incidents near the West Bank cities of Nablus and Hebron - the sort of death toll reminiscent of previous uprisings against Israel's prolonged military rule there.
The violence has sparked protests outside the region.
Demonstrators in London marched from the Israeli embassy to the Houses of Parliament and Whitehall, blocking traffic throughout the West End. French police clashed with pro-Palestinian protesters who defied a ban by authorities to march in central Paris.
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