United Nations Secretary-General Ban Ki-moon has urged Israel and the Palestinians to "stop fighting" and "start talking" to end the conflict in Gaza.
He was speaking in Israel as diplomatic efforts intensified.
More than 600 Palestinians and 30 Israelis have been killed in the past 14 days of fighting, officials say.
Earlier, US Secretary of State John Kerry said an Egyptian plan could form the basis of a ceasefire.
Speaking in Egypt, Kerry said the US was concerned about Palestinian casualties, but lent his support to Israel's "appropriate and legitimate" military operation.
He also said the US was sending $47m (£28m) in aid to Gaza "to alleviate the immediate humanitarian crisis".
Kerry is expected to stay in Cairo until at least Wednesday for talks with Egyptian officials and the Arab League chief Nabil al-Arabi.
At a joint news conference with Israeli Prime Minister Benjamin Netanyahu, Ban urged Israel to exercise "maximum restraint", adding that "military action will not increase Israeli security in the longer term".
He called on the Palestinians to pursue a policy of "no violence, recognition of Israel and respect for previous agreements".
Replying to Ban's opening comments, Netanyahu asked: "What grievance can we resolve for Hamas? Their grievance is that we exist."
— The Straits Times (@STcom) July 22, 2014
The latest Palestinian death toll of more than 600 was announced by Gaza's health ministry, which also said that 3,640 people had been injured.
The UN relief agency in Gaza says more than 100,000 Palestinians have taken refuge in its schools. It says 43% of Gaza has been affected by evacuation warnings or declared no-go zones.
BBC's Yolande Knell reports from Gaza City
Some 50 Palestinians have been killed here since midnight local time. Health officials told us that doctors tried to rescue the baby of a pregnant woman from Jabaliya who was killed in an air strike, only for the child to die.
Close to the BBC office in Gaza City, body parts could be seen in the rubble of the "Peace Tower" apartment building. Two families - 11 people - were killed overnight.
At the Rafah border crossing a BBC team was told only two injured people were able to exit to Egypt, leaving a frustrated crowd behind them.
Most ordinary Palestinians are weary and afraid. They sorely want an end to this fighting but many also insist conditions for a ceasefire must be met - including easing the tight border restrictions imposed on Gaza by Israel and Egypt.
BBC's Chris Morris reports from Jerusalem
The vast majority of the people killed in this conflict have been Palestinian civilians in Gaza. But Israeli military casualties are unexpectedly high - 28 soldiers killed so far.
If anything, this has increased a sense of solidarity. Military service is compulsory for most here, and there is huge sympathy for the armed forces.
One of the soldiers killed in Gaza on Sunday, Sean Carmeli, held joint US-Israeli citizenship and his family lives in the United States.
So the Israeli football team he supported, Maccabi Haifa, appealed on Facebook for fans to attend his funeral, to ensure he was not buried alone. Twenty thousand people turned up.
Plenty of Israelis are uncomfortable with Gaza's high civilian death toll. But a clear majority continues to support a military operation designed to limit Hamas's ability to fire rockets into Israel, and to infiltrate Israeli territory.
The extent of the cross-border tunnel network dug by Hamas has been a big shock for some. And Israelis want the threat removed.
— Mikko Takkunen (@photojournalism) July 20, 2014
The majority of Palestinians killed were civilians, including dozens of children, according to the UN.
The IDF also says it has killed at least 170 militants. Israel says 28 of its soldiers and two Israeli civilians have died over the past two weeks.
— Wall Street Journal (@WSJ) July 22, 2014
Early on Tuesday, Israeli air strikes destroyed several mosques and targeted a stadium and the home of the late leader of Hamas's military wing.
On Monday the UN Security Council has issued a call for an "immediate cessation of hostilities", but did not endorse a strongly worded draft resolution backed by Arab states.
— Agence France-Presse (@AFP) July 22, 2014
Israel launched its ground operation in Gaza after days of air strikes, following rocket fire by militants into Israeli towns.
It says the move is necessary to target Hamas' network of tunnels, which have been used by militants to get into Israel and carry out attacks.
— POLITICO (@politico) July 22, 2014
Meanwhile, Israel says a soldier hit in a Hamas attack in Gaza on Sunday is missing. Media say he is presumed dead.
Grad Rocket Launchers Discovered Next to Gaza School http://t.co/21QVVgbf4C
— IDF (@IDFSpokesperson) July 21, 2014
On Tuesday, the IDF named 21-year-old Sgt Oron Shaul as the last of seven soldiers reported killed in a single incident in Shejaiya, near Gaza City, on Sunday.
July 20.The Golani Brigade enters Gaza on a mission: remove Hamas’ terrorist infrastructure & restore quiet to Israel http://t.co/UeIm2RQ89F
— IDF (@IDFSpokesperson) July 22, 2014
The troops are all believed to have died when their armoured personnel carrier was attacked.
Hamas said on Sunday evening that it had captured an Israeli soldier, but Israel's UN ambassador denied the claim.