Health officials and witnesses said Israeli forces shot dead 21-year-old Razan al-Najar, a volunteer medic, as she ran towards the border fence, east of the south Gaza city of Khan Younis, in a bid to reach a casualty.
The Israeli military said Palestinian militants had attacked its troops along the border with gunfire and a grenade.
In a written statement on Saturday, the military said it would investigate al-Najar's death.
Thousands of people attended al-Najar's funeral in Gaza on Saturday, including some she had treated when they were wounded at previous border protests. Her body was wrapped in a Palestinian flag and carried through the streets on a stretcher by mourners.
"With our souls and blood we redeem you martyr Razan," cried mourners as the body was brought to her home for a last farewell before burial.
Residents said al-Najar was a popular figure at the protest sites and pictures depicting her as an angel circulated on Palestinian social media.
Her death brought to 119 the number of Palestinians killed in weekly demonstrations launched on March 30 in the Gaza Strip.
The UN Middle East peace envoy Nickolay Mladenov condemned al-Najar's death in a statement posted to his Twitter account.
Israel has drawn international condemnation for its use of deadly force during the mass demonstrations. It says many of those killed were Hamas members and militants trying to launch attacks under cover of the protests.
The Palestinians say most of the dead and the thousands wounded were unarmed civilians against whom Israel was using excessive force.
In a statement published by the Palestinian official news Agency Wafa on Friday, Palestinian Health Minister Jawad Awwad condemned al-Najar's killing and said it violated international law.
The Israeli military said in its statement: "The IDF constantly works to draw operational lessons and reduce the number of casualties in the area of the Gaza Strip security fence. Unfortunately, the Hamas terror organization deliberately and methodically places civilians in danger."
On Friday the United States, Israel's ally, vetoed a Kuwaiti-drafted UN Security Council resolution that condemned Israel's use of force against Palestinian civilians.
Later, a second, US-drafted resolution that blamed Hamas for the violence and upheld Israel's right to defend itself failed to attract any other country's support when it was put to vote in the 15-member council.