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 was “killed by Israeli occupation soldiers in the city of Nablus”.
The White House did not ascribe blame, but called on Israel to investigate.
Earlier, US State Department spokesman Matthew Millersaid Washington is “urgently gathering more information about the circumstances of her death”.
Ms Eygi was born in Antalya, as reported by Turkish media.
The dual-nationalwas 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 died from a “gunshot in the head”.
In a statement, the IDF said: “Today (Friday), during Israeli security forces activity adjacent to the area of Beita, the forces responded with fire toward a main instigator of violent activity who hurled rocks at the forces and posed a threat to them.
“The IDF is looking into reports that a foreign national was killed as a result of shots fired in the area. The details of the incident and the circumstances in which she was hit are under review.”
Jonathan Pollak, an Israeli activist who was at the same protest as Ms Eygi, said he heard “two separate shots of live ammunition, shot one after the other… and then I heard another shot”.
“I found her lying on the ground, beside the tree, bleeding from her head,” Mr Pollak told AFP news agency, as he showed blood on the hand he used to stop Ms Eygi’s head from bleeding.
“I took her pulse, she had a very weak pulse, we called the ambulance.
“From there we evacuated her to the village’s medical centre, where the doctor came into the ambulance and continued into the hospital, where they tried to resuscitate her but failed.”
Mr Pollak said the killing was “now receiving the light of day because she is an American citizen”.
In an interview with the BBC, Mr Pollak was asked about the IDF’s statement, where the Israeli military said security forces were responding to “a main instigator who hurled rocks at the forces and posed a threat to them”. Mr Pollak said there were clashes, but felt that “soldiers were under no threat”.
He also said that where Ms Eygi was shot was “a separate incident at a separate place,” adding that there was “no stone throwing” where she was.
It comes as Israeli forces withdrew from Jenin city and its refugee camp in the occupied West Bank on Friday, following a major nine-day operation there.
The Palestinian health ministry says at least 36 Palestinians were killed – 21 from Jenin governorate – in that time. Most of the dead have been claimed by armed groups as members, but the ministry says children are also among those killed.
In the past 50 years, Israel has built settlements in the West Bank and East Jerusalem, where more than 700,000 Jews now live.
Settlements are held to be illegal under international law – that is the position of the UN Security Council and the UK government, among others – although Israel rejects this.
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