"We firmly condemn this new Zionist crime against our people who were demonstrating peacefully," the group, which controls the Gaza Strip and whose leader Khaled Meshaal is based in Damascus, said in a statement.
"We call on the international community to assume its responsibility in light of the continuous Zionist crimes against our people," it said, adding that Sunday's clashes marked a new chapter in the struggle against the "Zionist occupation."
"We say to the entire world that the return of the Palestinian refugees to their homeland will remain a central part of the Palestinian question," the statement said.
Twelve people were killed and around 300 others injured on Sunday when Israeli troops fired on thousands of people along the Syrian and Lebanese borders, as well on the Gaza Strip and the West Bank.
The protests came as Palestinians in the territories and neighbouring countries staged massive protests to mark the anniversary of Israel's founding in 1948 in an event known in Arabic as the "Nakba" or "catastrophe".