The health ministry confirmed Mahmoud al-Masri, 30, was killed in clashes along the Israeli-Gaza border in Khan Yunis in southern Gaza.
The Israeli army confirmed it had shot two people along the border, accusing them of being "main instigators" of "violent riots."
Trump's Wednesday announcement that he would move the embassy and recognise Jerusalem as Israel's capital sparked anger across the Palestinian territories and the wider Muslim world.
Gaza's Islamist rulers Hamas had called for a day of rage Friday.
Clashes were reported in a number of locations along the Gaza border fence, as well as in the West Bank and Jerusalem.
More than 250 Palestinians were injured, mainly from tear gas, according to the health ministry.
The Israeli army said "violent riots" had broken out in around 30 locations in the West Bank and Gaza Strip.
The West Bank clashes involved some 3,000 Palestinians, while hundreds of others were involved in the clashes on the Gaza border, it said.
The UN Security Council was to meet later Friday in an emergency session to discuss Trump's move, which has drawn near universal condemnation, including from United Nations Secretary-General Antonio Guterres.
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