Israel and the Palestinian authority Hamas have agreed to restore calm after a day of armed confrontation, which saw one Israeli soldier killed and dozens of airstrikes launched by Israel, killing four Palestinians.
The UN's special coordinator for the Middle East peace process, Nickolay Mladenov, wrote on Twitter: "Everyone in Gaza needs to step back from the brink".
Israel has said it is defending its sovereign border.
The past week had seen efforts to try to keep the situation under control, but Israel's Defence Minister Avigdor Lieberman warned on Friday of a "much tougher" response against Hamas if continued to fire rockets from the Gaza Strip.
The soldier was shot in a military operation near southern Gaza and later died from his injuries.
Israel has also further tightened its blockade of Gaza by restricting use of the only goods crossing, suspending oil and gas deliveries.More news: CW Developing 'Batwoman' Show With Lesbian Lead
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The Israeli military said it had launched a "wide-scale" attack on Gaza on Friday after one of its soldiers was shot dead near the border fence of the Palestinian territory, stoking fears of a new conflict.
Mr. Lieberman said Israel is in discussions about the violence in Gaza through intermediaries including Egypt and the United Nations but isn't speaking directly with members of the militant group.
Palestinians are calling to return to the homes their families were forced from in 1948, during the war surrounding the creation of Israel. Right NOW! Those who want to provoke #Palestinians and #Israelis into another war must not succeed.
Israel resumed air attacks on Gaza after last week's projectile attacks, warning that it would take whatever action necessary to stop incendiary kites and balloons being flown over the border. But more recently, some Gaza officials said the protests would end if Israel lifted an economic blockade of the Gaza Strip.
Khalil al-Hayya, a top Hamas leader, said "the protests will remain, flaming and existing, and its tools will multiply and diversify, including kites, until our goals are achieved". He said they would be returned only as part of a deal similar to that in 2011 under which captured Israeli soldier Gilad Shalit was released in exchange for more than 1,000 Palestinians jailed in Israel.