* Abbas condemns "massacre"; death toll from conflict tops 400
* 13 Israeli soldiers killed - biggest daily toll in years
* Israel says area housed rockets, locals warned to evacuate
* U.N.'s Ban to see Abbas in Qatar, tour region in truce bid (Adds Netanyahu comments, U.N. figures on displaced in Gaza)
By Nidal al-Mughrabi and Jeffrey Heller
GAZA/JERUSALEM, July 20 (Reuters) - More than 60 Palestinians and 13 Israeli soldiers were killed as Israel shelled a Gaza neighbourhood and battled militants on Sunday in the bloodiest fighting in a near two-week-old offensive.
Palestinian President Mahmoud Abbas accused Israel of carrying out a massacre in Shejaia in the eastern suburbs of the city of Gaza and declared three days of mourning.
Israel's army said it was targeting militants from Gaza's dominant Hamas group whom it alleged had fired rockets from Shejaia and built tunnels and command centres there. The army said it had warned locals two days earlier to leave.
It was the Israeli military's highest one-day death toll since a 2006 war against Hezbollah guerrillas in Lebanon.
Army sources said seven of the 13 soldiers were in an armoured personnel carrier hit by anti-tank fire. Others were killed setting up positions inside houses they had taken over, the sources added.
Residents fled Sunday's fighting along streets strewn with bodies and rubble, many of them taking shelter in Gaza's Shifa hospital.
Cries of "Did you see Ahmed?" "Did you see my wife?" echoed through the courtyard. Inside, dead and wounded lay on blood-stained floors.
Shifa hospital's director, Naser Tattar, said 17 children, 14 women and four elderly were among the 62 dead, and about 400 people were wounded in the Israeli assault.
Gaza's Health Ministry said 37 other Palestinians were also killed on Sunday, bringing the Gaza death toll since an Israeli air and naval bombardments began on July 8 to 433, many of them civilians. Some 2,600 Palestinians have been wounded, it said.
In all, 18 Israeli soldiers and two Israeli civilians have been killed since the launch of the offensive, which expanded on Thursday into a ground push, following mounting cross-border rocket attacks by militants.
Palestinian fighters kept up their rocket fire on Israel on Sunday. Sirens sounded in southern Israeli towns and in the Tel Aviv metropolitan area. There were no reports of casualties from those salvoes.
Israeli Prime Minister Benjamin Netanyahu said he had spoken again with U.S. President Barack Obama about the Gaza situation on Sunday.
At a news conference in Tel Aviv, after the military released news of the soldiers' deaths, Netanyahu pledged to press on with the campaign.
"We are undeterred. We shall continue the operation as long as is required," he said.
DEATH TOLL TOPS 400
Thousands streamed out of Shejaia, some by foot and others piling into the backs of trucks and sitting on the hoods of cars filled with families trying to get away. Several people rode out of the neighbourhood in the shovel of a bulldozer.
Video given to Reuters by a local showed at least a dozen corpses, including three children, lying in streets, though the footage could not be verified independently.
As the tank shells began to land, Shejaia residents called radio stations pleading for evacuation. An air strike on the Shejaia home of Khalil al-Hayya, a senior Hamas official, killed his son, daughter-in-law and two grandchildren, hospital officials said.
Hamas's armed wing, the Izz el-Deen al-Qassam Brigades, said it used landmines and roadside bombs against advancing Israeli tanks and armoured personnel carriers.
"As we moved into Shejaia we were met by anti-tank missiles, RPGs, heavy, extensive weapons fire at the forces from the houses, from the surrounding buildings," said Lieutenant-Colonel Peter Lerner, an Israeli military spokesman.
Israeli Defence Minister Moshe Yaalon said some of the Israeli firepower was unleashed to try to extricate soldiers who had come under attack.
The Israeli military said it beefed up its presence on Sunday, with a focus on destroying missile stockpiles and the vast tunnel system Hamas built along the frontier with Israel.
"I estimate that within two or three days the major part of the tunnels will have been destroyed," Yaalon said at the news conference where Netanyahu also spoke.
Netanyahu, who has accused Hamas of using non-combatants as human shields, told CNN the army was concentrating on military targets.
"All civilian casualties are unintended by us, but actually intended by Hamas. They want to pile up as many civilian dead as they can it's gruesome," Netanyahu said. "They use telegenically dead Palestinians for their cause. They want the more dead the better."
The U.N. Relief and Works Agency said that 81,000 displaced people had now taken refuge in 61 UNRWA shelters in Gaza.
Egypt, Qatar, France and the United Nations, among others, have all been pushing, with little sign of progress, for a permanent ceasefire in the worst surge of Israeli-Palestinian fighting in two years.
U.S. Secretary of State John Kerry said he might travel to the Middle East soon to try to aid truce efforts. He said he supported Israel's efforts to destroy tunnels it says Gaza militants use for infiltration attempts and to hide weaponry.
"We support Israel's right to defend itself against rockets that are continuing to come in," Kerry told Fox News.
Washington has also urged Israel to minimise civilian casualties.
Qatar was due to host a meeting between Abbas and U.N. Secretary-General Ban Ki-moon on Sunday, a senior Qatari source told Reuters. Ban was due to travel to Kuwait, Egypt, Israel, the Palestinian Territories and Jordan during the week, a U.N. statement said.
The Qatari source said Abbas would also meet Hamas leader Khaled Meshaal.
Western-backed Abbas in April struck a reconciliation deal with Hamas, which seized the Gaza Strip in 2007 from forces loyal to his Fatah movement. The agreement led to the formation of a Palestinian unity government and Israel's pullout from U.S.-brokered peace talks.
Hamas has already rejected one Egyptian-brokered truce, saying any deal must include an end to a blockade of the coastal area and a recommitment to a ceasefire reached after an eight-day war in Gaza in 2012.
Hostilities escalated following the killing last month of three Jewish students that Israel blames on Hamas. Hamas neither confirmed nor denied involvement.
The apparent revenge murder of a Palestinian youth in Jerusalem, for which Israel has charged three Israelis, further fuelled tension. (Additional reporting by Noah Browing in Gaza and Ari Rabinovitch in Jerusalem; Writing by Jeffrey Heller)