Prisoner Yerden Bibas, whose wife Sherry and two children Kfir and Ariel were killed during Israeli raids on the Gaza Strip, told the Israeli prime minister: "Bibi (Benjamin Netanyahu) bombed and killed my wife and two children, who were the most important thing in my life. Please return their bodies for burial in Israel.
Hamas said it "offered to hand over the three bodies, but the Israeli government refused to receive them and is still manoeuvring and bargaining," according to its statement at the beginning of the video.
In a televised press conference, IDF spokesman Daniel Hagari described the footage as a "terrible and cruel" form of psychological warfare waged by the Palestinian faction.
Hamas announced on Wednesday evening that the family members had been killed during Israeli shelling that preceded the truce that began last Friday.
The group said dozens of detainees were killed by Israeli shelling during the days of war, with some estimates numbering about 60.