Funerals began on Tuesday for some of the 58 people killed, coinciding with the day Palestinians call the "Nakba", or "catastrophe", which commemorates more than 700,000 Palestinians who fled or were expelled from their homes in the 1948-49 war surrounding Israel's creation.
The opening of the USA embassy coincided with the 70th anniversary of Israel's founding.
Netanyahu defended his military, saying: "Every country has an obligation to defend its borders".
A 1948 United Nations resolution supporting the right of return has never been implemented.
Palestinian supporters gathered on the steps of the U.S. Embassy on Sussex Drive to send a message to U.S. President Donald Trump - and to other leaders around the world - that the decision to allow the Americans to move their embassy from Tel Aviv to Jerusalem contravenes global law.
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The move came as a blow to Palestinians, who viewed part of Jerusalem as capital for a possible future state. Trapped by a brutally enforced sea, land and air blockade, and caught in a power struggle between Hamas and the Palestinian Authority, they live with limited electric power, little access to safe water, poor medical care and awful poverty.
The protests peak on Fridays, with the numbers of demonstrators ranging from thousands to tens of thousands. Some have tried to break through the fence.
Earlier, the Israeli army air-dropped leaflets over Gaza warning people not to approach the fence that separates Gaza from Israel. Tuesday's statement regarding the violence on the border with Gaza differs from previous statements on protests at the Gaza border by specifically mentioning the terror group Hamas. No Israeli casualties have been reported.
Palestinian protests are also planned for today. The soldiers' response, it said, was in accordance with "standard operating procedures".
Palestinian Health Ministry officials said 55 protesters were killed and more than 2,700 injured either by live gunfire, tear gas or other means.
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What has been the worldwide reaction?
UN Secretary-General Antonio Guterres said he was "profoundly alarmed and concerned by the sharp escalation of violence" and the number of dead. May defended Israel's right to defend itself, but also called for greater restraint to be exercised.
"The IDF (Israel Defence Forces) will act forcefully against any terrorist activity and will operate to prevent attacks against Israelis", the military said in a statement.
But the South African Jewish Board of Deputies and the SA Zionist Federation were unhappy and said in a statement on Monday that the government's decision to withdraw the ambassador smacked of gross double-standards against Israel.
Ireland summoned the Israeli ambassador to express its "shock and dismay at the level of death and injury".
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"Every casualty on the border is a victim of Hamas' war crimes, every death is a result of Hamas' terror activity, and these casualties are exclusively Hamas' responsibility", Danon said in a statement.
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