In his visit, Gallant warned that "those who think we are delaying [the invasion of Rafah] will soon see that there is no place we cannot reach."
Michael Fakhri claimed US aid to Gaza is a "performance."
Prime Minister Benjamin Netanyahu had reportedly postponed a decision on the issue, with civil order increasingly strained in Hamas-run Gaza.
Under the arrangement, cargo would undergo security inspections on Cyprus, by a team including Israeli officials.
The Fact Finding and Assessment Mechanism (FFAM) is expected to conduct an independent investigation.
A convoy of trucks started making its way towards Gaza City, but struggled to make progress as hungry crowds tried to attack the trucks, which then faced gunfire as they entered the city.
Hundreds of people were seen running towards the trucks as they crossed into Gaza from Egypt, and several videos showed Gazans climbing up and throwing aid to those on the ground.
"The Commission supports Israel's right to defend itself against the Hamas terrorists, in full respect of international humanitarian law," the EU executive said in a statement.
Qatar halted periodic payments of some $7 million from the Gulf to Gaza for the civil servants' salaries some three months ago, as per a local report.
Many people in surrounding villages say they are surviving off donations from local charities and have received little or no help from the central state authorities.
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