(Adds Obama comments)
By Nidal al-Mughrabi and Crispian Balmer
GAZA/JERUSALEM, July 21 (Reuters) - The Palestinian death toll in an Israeli offensive in the Gaza Strip jumped to more than 500 on Monday, as the United States, alarmed by escalating civilian bloodshed, took a direct role in efforts to secure a ceasefire.
Despite growing calls for a halt to two weeks of fighting, violence raged on, with Israel saying it had killed 10 militants who tunnelled across the border from Gaza, and Palestinian officials accusing the Israeli army of shelling a hospital.
Israeli jets, tanks and artillery constantly pounded the densely-populated coastal strip, killing 28 members of a single family at the southern end. Hamas unleashed regular volleys of rockets at Israeli cities, many of them intercepted.
A day after he was caught by an open microphone saying sarcastically that the Israeli assault was "a hell of a pinpoint operation", U.S. Secretary of State John Kerry flew to Cairo to try to secure an end to hostilities.
Speaking in Washington, President Barack Obama said he was increasingly worried by the conflict.
"We have serious concerns about the rising number of Palestinian civilian deaths and the loss of Israeli lives, and that is why it now has to be our focus and the focus of the international community to bring about a ceasefire," he told reporters at the White House.
The Islamist group Hamas, which inflicted the biggest single loss on Israeli forces in eight years when it killed 13 soldiers in Gaza on Sunday, said it would not lay down its arms until a series of demands were met -- including an end to a blockade imposed on the territory by both Israel and Egypt.
"The world must understand that Gaza has decided to end the blockade by its blood and its heroism," deputy Hamas leader Ismail Haniyeh said in a televised address.
At Al-Aqsa hospital in the central Gaza Strip, four people were killed and 70 wounded when an Israeli tank shell slammed into the third floor, housing operating theatres and an intensive care unit, the Health Ministry said.
The Israeli military, which has accused Hamas militants of firing rockets from the grounds of Gaza hospitals and seeking refuge there, had no immediate comment.
Non-stop attacks lifted the Palestinian death toll to 518, including almost 100 children, since fighting started on July 8, Gaza health officials said. Israel says 18 of its soldiers have also died along with two civilians.
Hamas announced late on Sunday it had captured an Israeli soldier in Gaza, displaying a photo ID card and serial number, but no image of the man in its hands. The Israeli army said it was still investigating the allegation.
The Hamas announcement set off rejoicing in the embattled Gaza Strip.
"This is not the time to talk of a ceasefire," said Gilad Erdan, communications minister and a member of Israeli Prime Minister Benjamin Netanyahu's inner security cabinet.
"We must complete the mission, and the mission cannot end until the threat of the tunnels is removed," he told reporters.
Looking to take the fight onto Israeli soil, two groups of Palestinian fighters crossed from Gaza via two tunnels in the early morning, opening fire as they entered.
Black and white surveillance footage supplied by the army, showed one group of five or six men crouching and firing in long grass. Seconds later they were hit by a large explosion, which sent a cloud of smoke and debris flying into the air.
A military spokeswoman said at least 10 militants died. She did not comment on reports of casualties amongst Israeli forces. Hamas said its men had destroyed two army jeeps in the assault and said it had killed 10 Israeli soldiers in a separate ambush.
Fighters from Hamas, which controls Gaza, and its allies, have repeatedly tried to infiltrate Israel over the past week through a vast network of hidden tunnels, looking to attack villages and army encampments that dot the border area.
Netanyahu sent in Israeli ground forces on Thursday to destroy the tunnels and the militants' missile stock pile.
A confident-sounding Hamas told Israel to quit Gaza.
"Israel has terribly failed and we advise them to take their soldiers and leave before we kidnap more soldiers in addition to the scores we have already killed and wounded," said the group's spokesman, Sami Abu Zuhri.
Any capture of an Israeli soldier would pile pressure on Netanyahu to intensify the military campaign.
He agreed to free more than 1,000 Palestinian prisoners in 2011 to secure the release of a soldier who was held for more than five years by Gaza, and officials have said they want to avoid any repeat of that prolonged drama.
Lerner told reporters that the main focus of fighting remained the Shejaia district, east of Gaza City, where some 72 Palestinians, many of them civilians, were killed on Sunday.
In its push against militants in Shejaia, Israel also suffered its worst losses in the offensive, with 13 soldiers killed - the army's heaviest one-day loss in battle since 2006.
The carnage energised world leaders to step up efforts to find a way out of the confrontation but a rift among Arab powers may complicate the quest for a truce.
Past conflicts between Israel and its foes in Gaza and Lebanon have usually ended when the United States, the Jewish state's guardian ally, calls a halt, sometimes hastened by a strike that inflicts high civilian casualties on the Arab side.
While Washington has called for calm, it has so far defended Israeli actions and has not pressured Netanyahu publicly.
The high death toll by Israeli standards appeared to cement a public mood of grim determination.
Many flags flew at half mast but no leading figures are calling into question the operation. "We need to continue to grit our teeth, to shut our ears, to ignore the background noise and to get the job done," columnist Ben Caspit wrote in Ma'ariv.
Israel's army said it had been targeting militants in the clashes, charging that they had fired rockets from Shejaia and built tunnels and command centres there. The army said it had warned civilians to leave two days earlier.
Sounds of explosions rocked Gaza City through the morning, with residents reporting heavy fighting in Shejaia and the adjacent Zeitoun neighbourhood. Locals also said there was heavy shelling in Beit Hanoun, in the northern Gaza Strip.
"It seems we are heading towards a massacre in Beit Hanoun. They drove us out of our houses with their fire. We carried our kids and ran away," said Abu Ahmed, he did not want to give his full name for fear of Israeli reprisals.
At the other end of Gaza, medics said 28 members of the Abu Jamea family died when their house was hit by a bomb. Nearby, 10 members of the Seyam family died when they were hit by a tank shell as they tried to flee their house, officials said.
(Additional reporting by Ari Rabinovitch in Jerusalem, Noah Browning in Gaza and Michelle Nichols at the United Nations; Writing by Crispian Balmer and Allyn Fisher-Ilan; Editing by John Stonestreet and Paul Taylor)