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The crossing has been the only point of entry for humanitarian relief to get into the Palestinian enclave after Israel imposed a complete blockade in response to the October 7 Hamas attack and embarked on a series of devastating air strikes.
Since limited aid deliveries resumed through the Rafah crossing on October 21, a total of 117 lorries have entered Gaza.
Before the siege, about 500 lorries carrying aid and other goods entered the Palestinian territory daily.
“Impeding relief supplies as provided by the Geneva Conventions may constitute a crime within the court jurisdiction,” Mr Khan said on Sunday evening.
“There should not be any impediment to humanitarian relief supplies going to civilians. They are innocent, they have rights under international laws.”
Mr Khan said he saw lorries full of goods and humanitarian assistance “stuck where nobody needs them, stuck in Egypt, stuck at Rafah”.
He said supplies must get to civilians as soon as possible without any hindrance.
The ICC is carrying out an investigation into crimes committed in the Palestinian territories – “whether it’s by Israel or Palestine or whether it’s acts committed on the territory of Palestine or from Palestine into Israel”.
Mr Khan said he hoped to visit Gaza and Israel in the coming days to underline clearly to Israel that “there must be discernible efforts, without further delay”, to make sure civilians in the Hamas-run territory receive basic food and medicines.
“These are the most tragic of days as the suffering of children and women and men is profound,” he said.
Mr Khan said he was concerned about the increase of reported deaths and injuries in Gaza, the occupied West Bank and Israel.
“This must be a moment where we share our humanity and find the common ground together,” he said.
Israel has said 1,400 people, mostly civilians, were killed in the Hamas attacks that set off the war. At least 220 were taken hostage.
Gaza's Health Ministry says more than 8,000 people, including thousands of women and children, have been killed in Israeli air and ground attacks since.
Mr Khan said taking hostages was in breach of the Geneva Conventions.
“I call for the immediate release of all hostages taken from Israel and for their safe return to their families,” he said.
Israel has moral and legal obligations to comply with the laws of conflict, he said.
“We need to separate allegations from facts,” he said, as his team is independently looking into the events in Palestine and Israel.
The ICC has been investigating the actions of both Israeli and Palestinian authorities since 2014.
Mr Khan said the world must reflect on a moment of objectivity and reflection for the world “built on the rubble of World War II, the terrible gas chambers and the Holocaust, the razing of cities throughout Europe, [which] was meant to create institutions that would ensure never again would we see abominations”.