The ministry did not immediately identify the victims, but said they had been killed as thousands of protesters approached the heavily-guarded Israeli border.
CNN crews witnessed Palestinians throwing grenades and burning tires, and burning the border fence - with Israeli forces responding with tear gas, live fire and a water cannon.
Abbas says that arranging deals with Hamas is equivalent to recognizing its control over Gaza.
Israeli army spokesman Jonathan Conricus said on Twitter an "organised attack" had involved around 20 Palestinians crossing the border after an explosive device destroyed a portion of the fence.
Gaza medics said that, in addition to the six dead, around 140 others were wounded.
Israeli police said on Saturday they were investigating the death of a Palestinian woman in the occupied West Bank, after her husband said he suspected Israeli settlers had pelted their vehicle with rocks.
According to the Israeli army, rioters breached the security fence and planted a bomb nearby, setting a section of the barrier alight.More news: Midterm elections in wake of Kavanaugh confirmation
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Four Israeli tanker trucks had delivered fuel to Gaza on Friday morning before "riots" broke out, Lieberman said.
The deaths occurred during Friday protests, which often turn violent, along the fence separating Israel and the Gaza Strip.
The Israeli army said approximately 14,000 people had gathered at various sites along the border.
Although the fuel was aimed at making the situation better in Gaza- where residents are now only receiving around four hours of electricity per day on average - the Qatar-bought delivery was protested by Palestinian President Mahmoud Abbas, as it was arranged with Hamas rather than his government.
A group of Israeli paramilitary settlers attacked a Palestinian couple south of Nablus, in the northern part of the West Bank, on Friday, killing the woman and severely injuring her husband.
Israel accuses the enclave's rulers Hamas of leading the protests and using them as a cover for attacks.
More than 150 Palestinians have been killed since the marches began.