Gaza City (CNN) -- Hamas claimed it captured an Israeli soldier Sunday on the deadliest day yet in the battle between Israel and Hamas militants in Gaza.

Ron Prosor, Israeli ambassador to the United Nations, denied the report.

"There's no kidnapped Israeli soldier, and those rumors are untrue," he said.

According to Abu Obeida, a spokesman for Hamas' military wing, al-Qassam Brigades, the soldier was taken during an early morning operation.

He provided the supposed soldier's name and ID.

"He is a prisoner, and if Zionists lie about the dead and wounded, then the fate of this soldier is their responsibility," the spokesman said.

Gunfire and cheers erupted in Gaza in apparent celebration of the soldier's capture, according to CNN reporters on the ground.

"It's a game changer, immediately, because it's going to change what the Israelis are doing on the ground in that sector. They're going to be looking for him," said CNN military analyst Lt. Col. Rick Francona.

"Overall, the Israeli strategy is not going to change. They're committed to this mission," he said. Though, in the future, Francona said, the Israelis are "going to have to make some sort of accommodation to get this guy back."

In 2006, Israeli soldier Gilad Shalit was captured. He was released some five years later in exchange for more than a thousand Palestinian prisoners.

Gaza battle's deadliest day for both sides

Eighty-seven Palestinians died, at least 60 of them in Israel's assault on the town of Shaja'ia, the Gaza Health Ministry said.

The IDF said 13 soldiers were killed. At a news conference, Israeli Prime Minister Benjamin Netanyahu expressed the country's "deep pain" at the loss of the soldiers.

Among those killed was Max Steinberg, a California native, according to the Jewish Federation of Greater Los Angeles. Steinberg attended Pierce College and served as a sniper.

Sean Carmeli, an IDF soldier from South Padre Island, Texas, was also killed, according to Rachel Simony of the Congregation Shoova Israel in South Padre Island.

In total, 476 Palestinians have been killed since the start of Israel's military operations against Hamas on July 8, according to the Gaza Health Ministry. It's unknown how many were militants. The United Nations has estimated that 70% were civilians.

The Health Ministry said 3,130 people have been wounded.

Since beginning ground operations Thursday, Israel said, it has killed at least 70 terrorists and captured others.

"We're doing everything we can not to harm the people of Gaza," Netanyahu added. "Hamas is doing everything they can to make sure the people of Gaza suffer."

But people in Gaza who spoke with CNN painted a different picture. "What is happening is a massacre," said a resident of the al-Remal neighborhood.

Hamas claimed responsibility for the deaths of the Israeli soldiers, saying it had lured tanks into a field in which it had hidden improvised explosive devices. The attack "destroyed the force completely," Hamas said, calling it a "heroic operation."

In total, 18 Israeli soldiers have been killed, in addition to two civilians. Israel has used its Iron Dome defense system to block many missiles, fired by militants in Gaza, from hitting population centers.

Dozens dead in Shaja'ia

Hundreds of people fled in panic into Gaza City on Sunday as Israeli troops focused their firepower on nearby Shaja'ia. Bodies lay in streets beside gashes blasted into apartment buildings, said people who had escaped the violence.

Overnight, Hamas fired rockets from Shaja'ia toward Israel.

For three days, the IDF had warned residents of Shaja'ia to flee, Israel said. Such warnings are delivered through calls and text messages as well as fliers that said "it is the intention of the IDF to carry out aerial strikes against terror sites and operatives" in the area. The fliers told people to head to Gaza City by Wednesday morning and not to return until further notice. The IDF posted an English translation of the fliers Sunday on Twitter.

Some residents said they received the warnings but felt that even if they fled, they could face the same dangers in other parts of Gaza.

But the IDF said Hamas "ordered them to stay" and "put them in the line of fire."

The IDF posted a photo Sunday on Twitter, saying, "We fired a warning shot at this target in Gaza. In response, these civilians ran to the roof and brought their kids."

Hamas' cease-fire demands

Hamas told CNN on Sunday that it would only agree to a cease-fire if it was guaranteed that certain demands would form the basis of negotiations. Izzat Risheq, a senior Hamas political leader in Qatar, said the demands include opening the border crossings, freeing detainees Israel arrested in June, and opening the Gaza port.

The militant group overnight turned down an invitation by Egypt to talk about a cease-fire initiative that Cairo had proposed.

U.S. Secretary of State John Kerry is scheduled to go to Egypt on Monday to meet with senior officials about the crisis in Gaza. While there, he will push for a cease-fire, a State Department spokeswoman said.

U.S. President Barack Obama spoke Sunday with Netanyahu, the second call in three days. Obama reiterated U.S. condemnation of Hamas attacks against Israel "and reaffirmed Israel's right to defend itself," the White House said in a statement. Obama also "raised serious concern about the growing number of casualties, including increasing Palestinian civilian deaths in Gaza and the loss of Israeli soldiers."

Speaking to CNN's "State of the Union," Kerry said the United States supports Egypt's initiative for a cease-fire and "will work for a fair cease-fire."

The United States has "shown our willingness to try to deal with the underlying issues," but Hamas "must step up and show a level of reasonableness," he said.

"Israel is under siege by a terrorist organization that has seen fit to dig tunnels and come through those tunnels with handcuffs and tranquilizer drugs, prepared to try to capture Israeli citizens and take them back to hold them hostage. No country could sit by and not take steps to try to deal with people who are sending thousands of rockets your way," Kerry said.

"No country, no human being, is comfortable with children being killed, with people being killed, but we're not comfortable with Israeli soldiers being killed either or with people being rocketed in Israel."

"Hamas uses civilians as shields," he said. "They fire from a home and draw the fire into the home."

Separately, Kerry was caught on an open mic, appearing to criticize Israeli assurances that its ground offensive in Gaza would be limited.

His comments occurred between multiple television interviews. He was heard in a phone conversation with a State Department deputy, Jonathan Finer, discussing the deaths of Israeli soldiers killed overnight.

"I hope they don't think that's an invitation to go do more," Kerry said in the unguarded moment. "That better be the warning to them."

At that point, Finer is heard mentioning the number of Palestinians wounded and killed in the past 24 hours.

"It's a hell of a pinpoint operation. It's a hell of a pinpoint operation," Kerry said, a seemingly frustrated comment aimed at Israel.

Fawzi Barhoum, a spokesman for Hamas, said on Aqsa TV on Saturday that there would be no truce or surrender while Israel is attacking.

Israel opens field hospital for Palestinians

Israel announced Sunday it would open a field hospital at the Erez Crossing to treat injured Palestinians. On Saturday, the defense forces delivered truckloads of medical supplies to Gaza.

Meanwhile, at Barzilai Hospital in Ashkelon, soldiers have been coming in with serious wounds from shrapnel and gunshots. The hospital treats soldiers and civilians, as well as injured Palestinians, although none were there Sunday.

The hospital is frequently hit by rocket attacks from Gaza. It has emergency procedures in place, including moving its neo-natal ward into a reinforced rocket shelter.

Israel agreed to a two-hour cease-fire Sunday, at the request of the Red Cross, to allow Palestinian emergency medical workers to tend to the wounded and dead in Shaja'ia, the IDF said. Israel also announced it was extending its cease-fire, but said Hamas was not holding its fire.

Hamas, meanwhile, said Israeli forces shelled Shaja'ia after the cease-fire was declared.

The IDF said it has held fire three times since beginning the operation in Gaza, but "Hamas never stopped shooting rockets."

Israel is still "early on in the mission," IDF spokesman Lt. Col. Peter Lerner said Sunday. "You can't erase 10,000 rockets overnight," he said of Hamas' arsenal.

The IDF is adding troops to the incursion. It called up tens of thousands of reservists at the start of Operation Protective Edge to prepare for the ground operations.

Israel said it has struck "2,300 terror targets" in Gaza and found 13 tunnels the militants use for smuggling weapons.

Netanyahu: Demilitarize Gaza

Netanyahu called on the international community to "undertake a program to demilitarize Gaza" in the future.

The situation is "unacceptable" because of Hamas and Islamic Jihad, Netanyahu told CNN's Wolf Blitzer in an interview Sunday.

"These people are the worst terrorists -- genocidal terrorists. They call for the destruction of Israel and they call for the killing of every Jew, wherever they can find them."

Hamas fighters in Gaza "don't care" about the dying people around them, Netanyahu said.

Israel has enabled the shipment of concrete into Gaza for buildings, hospitals, and schools, but the militants use hundreds of tons of it for each tunnel, Netanyahu said.

Hamas: Israel committed 'crime against humanity'

Hamas spokesman Sami Abu Zuhri, speaking to Al-Jazeera, said Israel committed "a crime against humanity," and that most of those killed in Shaja'ia were women and children. "Our people will not sit idle in front of this brutal aggression."

He called on the Palestinian Authority, in the West Bank, to "stop its security coordination with the occupation" and to "stop suppressing the demonstrations in the West Bank." He also said "the Arab world should not sit idle."

The Israeli government has repeatedly said that, unlike Palestinian militants, the IDF does not target civilians and works to avoid innocent casualties.

But in Gaza, one of the most densely populated areas on Earth, more than 70% of those killed in the hail of artillery and airstrikes have been civilians, according to the United Nations. A fifth were children. More than 40% of Gaza's population is 14 or younger.

About 81,000 Palestinians have taken refuge in U.N. facilities, Robert Turner, the director of U.N. efforts in Gaza, said Sunday. The United Nations has been investigating a cache of rockets used by militants found in a U.N. school.

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War-scarred Gaza medical crews also in harm's way

Opinion: A smart way out of the Gaza confrontation

CNN's Ben Wedeman and Karl Penhaul reported from Gaza; CNN's Ben Brumfield and Josh Levs from Atlanta. CNN's Atika Shubert reported from Israel near Gaza. CNN's Tim Lister, MIchael Martinez, Kareem Khadder, Ian Lee, Ali Younes, Mohammed Tawfeeq, Ralph Ellis, Samira Said, Michael Schwartz, Salma Abdelaziz and Tal Heinrich contributed to this report.