Gaza City (CNN) -- Angry clashes erupted Friday in the West Bank, as Palestinian leaders called for a "day of rage" in response to the fatal shelling of a United Nations school.
At least four Palestinians have been killed in outbreaks of violence in several parts of the West Bank, according to medical sources.
The call for a massive follow-up to what were already some of the largest West Bank protests in years comes as diplomats scrambled to find a cease-fire proposal that would satisfy mortal enemies Israel and Hamas and end more than two weeks of violence that has claimed more than 800 lives, most of them civilians.
Israeli police posted additional officers in Jerusalem ahead of Friday's protests, spokesman Micky Rosenfeld said on Twitter. Police had deployed between 3,000 and 4,000 officers to prepare for expected violence, the Jerusalem Post newspaper reported.
A 23-year-old man was shot near Huwara village outside Nablus by Jewish settlers, according to a doctor at the Rafidia Hospital. The circumstances of his death are unclear, but it led to clashes between protesters and the Israeli military in which another man was killed, medical sources said.
Two more men were killed during clashes with Israeli troops at a checkpoint north of Hebron in Beir Ummar, according to Palestinian medical sources.
A pro-Hamas demonstration in Ramallah drew a crowd of several hundred, and a protest march toward a Jewish settlement led to heavy clashes with IDF and border police.
Several people were injured, according to a CNN crew at the scene.
Elsewhere, an ambulance carrying wounded people in Gaza's Beit Hanoun district was struck by the Israeli military setting fire to the vehicle with all those inside, according to Dr. Ashraf al-Qidra, a spokesman for the Gaza Health ministry.
The Israel Defense Forces confirmed Friday that one of its soldiers, Sgt. Oron Shaul, was killed in battle in Gaza on Sunday.
Hamas had said it was holding Shaul after capturing him in an ambush on an armored personnel Sunday in which six other IDF soldiers died.
A committee led by the chief rabbi of the IDF concluded that he should be defined as "a soldier killed in action whose burial site is unknown," an IDF statement said.
Meanwhile, Secretary of State John Kerry was meeting with U.N. Secretary General Ban Ki-moon and Egyptian Foreign Minister Sameh Shukri Friday, and the Israeli Security Cabinet was meeting in closed session. But it was unclear if a deal to at least temporarily end the violence was near, or that Hamas would accept it.
Israeli newspaper Haaretz reported Friday that Kerry was hoping to announce a temporary cease-fire agreement, followed by negotiations between Israel and Hamas to reach a lasting end to the violence.
A senior State Department official speaking on background told reporters late Thursday that no deal had been reached but that discussions were ongoing.
Israeli government spokesman Mark Regev declined to comment on the security cabinet meeting, but said Israel is eager to find an end to the violence.
"We want to come out of this with a sustained period of quiet," he told CNN's Wolf Blitzer Friday.
"We don't just want a timeout for Hamas to recharge its batteries and to have more rockets on Israel."
West Bank protests
Thousands of Palestinians poured into the streets of the West Bank Thursday night to protest in support of Hamas and its conflict with Israel when Israeli security forces opened fire Thursday evening, Palestinian lawmaker Mustafa Barghouti said.
Protesters were "throwing rocks, firebombs and fireworks" at the troops, The Jerusalem Post reported. The newspaper said 13 Israeli police officers were injured.
Ramallah's health minister said 185 wounded Palestinians were brought in for treatment. At least two protesters were killed.
U.N. shelter hit
The demonstration erupted only hours after the U.N. shelter in Gaza was hit, killing 16 people and wounding a couple hundred more -- most of them women and children.
Video from the school showed chaos amid pools of blood. There were so many victims than many gurneys included two wounded children.
One father carried his small daughter into a hospital. There wasn't much he could do but try to comfort his little girl as she cried and begged for him not to leave her.
The bloodshed left U.N. Secretary-General Ban Ki-moon exasperated.
"I am telling to the parties -- both Israelis and Hamas, Palestinians -- that it is morally wrong to kill your own people," Ban said. The "whole world has been watching, is watching with great concern. You must stop fighting and enter into dialogue."
A Palestinian government statement called the shelling "Israeli brutal aggression."
It's unclear who was responsible. The Israel military said it is investigating.
"From initial inquiries done about the incident, during the intense fighting in the area, militants opened fire at ... soldiers from the school area," a military statement said. "In order to eliminate the threat posed to their lives, they responded with fire toward the origins of the shooting."
Regev said Friday that it could have been inadvertent Israeli fire or the result of a Hamas rocket strike on Israel that fell short.
"Even if we do discover in the end that it was errant Israeli fire, why was a U.N. school, the vicinity of a U.N. school, turned into a war zone by Hamas?" Regev said.
The Israeli military said the area surrounding the school in Beit Hanoun had turned into a battlefield, and it had asked that the facility be evacuated even before the school was hit. A four-hour window was given, the military said.
Chris Gunness, a spokesman for the United Nations Relief and Works Agency for Palestine Refugees in the Near East, disputed that Friday. He said the Israeli military never responded to the agency's urgent pleas for a cease-fire.
"If the IDF had responded, this carnage would never have happened," he said.
The Israeli military accused Hamas militants of refusing to let people at the shelter leave, saying they were being used as human shields.
"This is a tragedy. This is a clear tragedy," Israeli military spokesman Lt. Col. Peter Lerner told CNN.
Thursday's strike marks the third time a U.N. school serving as a shelter has been hit.
The first occurred Tuesday at an UNRWA school in eastern Gaza, where about 300 people were staying. The second occurred Wednesday in central Gaza at a shelter where about 1,500 were staying. There were no fatalities and few injuries in those incidents.
Diplomatic efforts continued into early Friday.
Kerry has been active, his shuttle diplomacy taking him from Cairo to Jerusalem, the Palestinian territories and back.
Asked Thursday about a possible cease-fire, he balked.
"I'm going to have a lot to say (Friday) probably, so I'm going to wait until then," Kerry said. "We still have more work to do. I certainly have more work to do tonight."
The diplomatic effort wasn't solely limited to the United States, as several Middle Eastern nations worked to try to win Hamas' agreement for an Egyptian-led cease-fire. Hamas said Turkey and Kuwait were also involved.
On the 18th day of the military operation in Gaza, Israeli Prime Minister Benjamin Netanyahu praised his country's military.
"The IDF (Israel Defense Forces) has struck a deep and significant blow at the infrastructures of terrorism -- at the terrorists themselves, at rockets, at command centers, at production facilities and at many other targets," he said Friday at the start of a Cabinet meeting.
But human rights organizations and the United Nations have argued civilians have been bearing the brunt of the violence in Gaza.
At least 826 people have been killed and more than 5,200 wounded since the start of an Israeli operation on Gaza, the Gaza Health Ministry said Friday.
An Israeli military representative said Friday that another soldier had been killed in Gaza, bringing the total number of Israelis killed to 36 -- 33 soldiers and three civilians.
"Both sides have violated international human rights, humanitarian law and human rights law," U.N. High Commissioner for Human Rights Navi Pillay said. "These are really shocking incidents."
Some flights to Israel resume
On Thursday, Delta Air Lines joined Air Canada and United Airlines in resuming flights to Ben Gurion International Airport in Tel Aviv. It had been a day and a half since the U.S. Federal Aviation Administration imposed the prohibition of flights to the city because of security concerns.
Lufthansa Group has canceled all Lufthansa, Germanwings, Austrian Airlines, Swiss and Brussels Airlines flights flying to and from Tel Aviv through Friday.