The latest Palestinian fatality was named as Shadi Shabat, 38, killed in clashes east of Gaza City. Another man was shot dead earlier east of Khan Yunis in southern Gaza, the ministry said.
The clashes come a week after similar demonstrations led to violence in which Israeli forces killed 19 Palestinians in the deadliest day since a 2014 war.
Israel's army has faced criticism over its use of live fire, but it has warned open-fire rules will remain unchanged to stop damage to the border fence, infiltrations and attacks.
On Friday, Israel's military said in a statement that "approximately 10,000 Palestinians have been rioting in five locations along the border with the Gaza Strip".
It added that "several attempts have been made to damage and cross the security fence under the cover of the smokescreen created by the burning tires that the rioters ignited".
The military also said explosive devices and firebombs were thrown.
"Troops are responding with riot dispersal means, including water cannons to put out fires, a huge fan to disperse the smoke, and fire in accordance with the rules of engagement," it said.
Protesters set alight mounds of tyres to create a smokescreen for Israeli snipers, sending thick black smoke into the air.
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