The draft statement would have expressed "outrage and sorrow" at the killings of palestinians, sought an "independent and transparent investigation."
Earlier Monday, the Palestinian U.N. envoy urged the council to condemn the killings.
Kuwait called for the session after 59 Palestinians were killed by Israeli gunfire.
Israeli gunfire killed 59 Palestinians and wounded over 2,700 in the Monday clashes -- the highest toll in a single day since a series of protests demanding the right to return to ancestral homes began on March 30.
The occupied territories have witnessed new tensions ever since US President Donald Trump on December 6, 2017 announced US recognition of al-Quds as Israel’s “capital” and said Washington would move US embassy to the city.
The dramatic decision triggered demonstrations in the occupied Palestinian territories as well as Iran, Turkey, Egypt, Jordan, Tunisia, Algeria, Iraq, Morocco and other Muslim countries.
