A young Palestinian was killed and 968 persons were injured as a result of Israeli repression against the occupation organized in the Gaza Strip for the third consecutive Friday.
Islam Herzallah, 28, died in hospital after being shot by Israeli troops east of Gaza City, the ministry said.
Israel says it is doing what it must to defend its border, and to stop any of the protesters getting across the fence.
Israel's army estimated there were 10,000 people "rioting" on Friday, with some attempting to breach the fence with firebombs and explosive devices.
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The April 13 riots featured protesters burning tires - which they had done the week prior - in addition to Israeli and American flags. We also urge Israeli soldiers to refrain from shooting live ammunition at unarmed Palestinian protesters from hundreds of meters away, across the fence separating the two territories.
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In total, 33 Palestinians have been killed and thousands wounded by Israeli forces since the "Great March of Return" began.
No Israelis have been killed during the demonstrations, and human rights groups say the Israeli military has used live fire against demonstrators who pose no immediate threat to life.
He said Hamas should not pursue a violent "struggle against Israel", but emphasised that Israel's "continuing control of Gaza's air, sea, and northern, southern, and eastern borders, and its restrictions on the freedom of movement of people, legitimate goods and equipment in and out of Gaza, have made the humanitarian situation worse".
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Several thousand people gathered Friday at one of the tent camps, east of Gaza City. Israel alleges Hamas is using protests as a cover for attacks.
The marches are expected to peak on May 15, the day after the 70th anniversary of Israel's declaration of independence but marked by the Palestinians as the Nakba Day, or "Day of the Catastrophe".
Yair Lapid, the leader of Israel's centrist Yesh Atid party, called Hamas a "despicable terror organization" and accused it of exploiting civilians.
Egypt, battling an insurgency in neighbouring Sinai, keeps its border with Gaza largely closed.
It added that the announced investigation was consistent with the approach of the Israeli military prosecution to blur the facts and give a false impression about the effectiveness of the Israeli law enforcement service.
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