The military had earlier said Gaza militants fired two rockets at Israel-one was shot down while the other fell short, with no immediate casualties reported.
On Saturday, the military said it thwarted a Palestinian attempt to damage the security fence surrounding Gaza and a group of militants briefly entered Israel before fleeing back into Gaza when Israeli troops opened fire.
Palestinian Health Minister Dr. Jawad Awaad said on Friday that Najjar was shot while clearly wearing a medical staff uniform, according to the Haaretz report.
It comes after a ceasefire was called on Wednesday.
No injuries were reported among Israeli soldiers.
The US proposal, meanwhile, said nothing about Israeli force or protecting Palestinian civilians, while casting blame on Hamas for violence, endangering civilians, and deteriorating conditions in Gaza.
Four separate attacks on southern Israel sent thousands of Israelis rushing to bomb shelters throughout the night. Israel's Iron Dome air defense system has intercepted the second missile launched out of Gaza on Israel on Sunday, the army press service said in a statement.
In a statement published by the Palestinian official news Agency Wafa on Friday, Health Minister Jawad Awwad condemned al-Najar's killing and said it violated global law.
Razan Al-Najjar, 21, a volunteering paramedic, was shot in the chest and killed by Israeli snipers while she was providing first aid to the wounded at the fortified border fence east of Khan Younes city.
Ambulances and medical crews attended the funeral, with her father holding the white blood-stained medics' jacket she wore when she was shot, as mourners called for revenge.
The person killed was identified as medical nurse Razan Ashraf Al Najjar, 20, ministry officials said.
In Israel's northern city of Haifa, meanwhile, Palestinian activists said they too planned to hold demonstrations in solidarity with their compatriots in blockaded Gaza.
The Kuwait-sponsored draft also "deplores" and demands a halt to "the use of any excessive, disproportionate and indiscriminate force" by the Israeli military.
Residents in Gaza said Israeli aircraft struck at least three sites belonging to Hamas, the Islamic terrorist group which controls the enclave.
"Hamas need to prevent incidents at the fence". It called for "an immediate global response to Israeli humanitarian law violations in Gaza". In response, the troops fired towards the terrorist, killing him.
Rocket sirens were sounded in southern Israel on Saturday evening, likely signalling a launch of projectiles from the Gaza Strip toward Israel.
Low-level demonstrations have continued since.
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