Reacting to the killing, Stéphane Dujarric, the spokesman for the UN secretary general, said: “We would want to see a full investigation of the circumstances and that people should be held accountable.”
Civilians, he added, “must be protected at all times”.
The US also called for an investigation into the incident. Sean Savett, spokesman for the White House’s National Security Council, said Washington was “deeply disturbed by the tragic death of an American citizen”.
“We have reached out to the government of Israel to ask for more information and request an investigation into the incident,” Mr Savett said.
Footage from the scene shortly after the shooting shows medics rushing Ms Eygi into an ambulance.
Jewish-Israeli activist Jonathan Pollak, who was at the protest, told BBC World Service’s Newshour programme he had seen “soldiers on the rooftop aiming”.
He said he had heard two separate shots, “with like a second or two distance between them”.
“I heard someone calling my name, saying in English, ‘Help us. We need help. We need help.” I ran towards them,” he said.
He said he had then seen Ms Eygi “lying on the ground underneath an olive tree, bleeding to death from her head”.
“I put my hand behind her back to try and stop the bleeding,” he said. “I looked up, there was a clear line of sight between the soldiers and where we were. I took her pulse, and it was very, very weak.”
He added that Friday’s demonstration had been Ms Eygi’s first time attending a protest with the International Solidarity Movement, a pro-Palestinian group.
The dual-national was rushed to a hospital in Nablus and later pronounced dead.
Dr Fouad Nafaa, head of Rafidia Hospital where Ms Eygi was admitted, confirmed that a US citizen in her mid-20s had died from a “gunshot in the head”.
US Secretary of State Antony Blinken deplored the “tragic loss”, while Turkish President Recep Tayyip Erdogan branded the Israeli action “barbaric”.
Turkey’s foreign ministry said Ms Eygi had been “killed by Israeli occupation soldiers in the city of Nablus”.
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