These groups accuse the Israeli military of not making enough effort to use other means of dispersing crowds.
Israeli officials say the soldiers are operating within international law against a mob led by the militant group Hamas that wants to break into Israel and carry out terrorist attacks.
About 1360 Palestinians were shot over the course of about eight hours on Tuesday, said the Palestinian Health Ministry in Gaza. All the dead were shot on the Palestinian side of the fence, and the border fence, though damaged, was never breached. No Israeli soldiers were reported injured.
Israel has drawn broad international criticism for allegedly using excessive force, facing questions about why protests by mostly unarmed Palestinians ended in such horrific bloodshed. Images and eyewitness accounts from the demonstrations appear at odds with Israel's insistence that its military response has been precise, carefully calibrated and intended to kill only as a last resort.
"Cutting or attacking the fence is an offence," said Michael Sfard, an Israeli human rights lawyer. "It has to be countered, but countered with reasonable force. There is no meter that I know of that would put the safety of the border fence at the same importance of the life of a 14-year-old."
Sfard is representing human rights groups petitioning Israel's Supreme Court to challenge the legality of the military's live fire rules during demonstrations in Gaza earlier this month. He said the only legal justification for using live ammunition against civilians is if they are "posing an imminent danger to the lives of others". In the state's response, Israel has argued that the protests can't be classified as civilian because they are part of the "armed conflict" between Israel and Hamas, which controls the Gaza Strip. "The state opposes the applying of human rights law during an armed conflict," Israel's response said.
The border is defended, in general, by two layers of fence topped with barbed wire, and Israeli snipers have been positioned within 100m of the barrier.
Lieutenant Colonel Jonathan Conricus, a spokesman for the Israeli military, said soldiers wouldn't be shooting to kill a fence-cutter like Khalil. He said it was a "hectic area with smoke and fires and lots of moving people". He added, "The command that is issued to our troops is to shoot towards the legs."
But multiple videos have surfaced apparently showing Israeli forces shooting unarmed protesters. A video in April showed a Palestinian running away from the border fence with a tyre before being shot in the head.
Israeli military officials say they have not changed their rules of engagement over nearly seven weeks of protests and warned before they started that there were orders to use "a lot of force" and live ammunition if soldiers or infrastructure came under threat. At least 111 Palestinians have been killed during the protests, Gaza health officials say.
Israeli officials have lavished praise on the actions of its forces in protecting the border. After a meeting with security chiefs on Tuesday, Prime Minister Benjamin Netanyahu's office released a statement saying that the "determined actions" of the Israeli miliary had "prevented a breach of Israel's borders".
The Israeli military reported an attempt to plant explosives along the fence and a shooting attack by eight Hamas militants in an armoured vehicle. In such cases, Israeli forces shoot to kill, Conricus said.
"The entire border riots are conducted under the slogan of the 'march of return'. What is the return? To annihilate Israel," said Yossi Kuperwasser, a retired brigadier general in the Israeli military who also served as director general of the Ministry of Strategic Affairs, voicing a common Israeli view.
If the military didn't prevent them from crossing, he said, "It would be a disaster for everyone."
Human rights groups say that fear doesn't justify the use of deadly force against unarmed protesters when they aren't posing an immediate threat.
