The crossing has been closed since 2011.
Ben-Gvir took aim at the military chief for the IDF's cooperation with Israel Police in foiling daily protests against the delivery of humanitarian aid into Gaza.
The protests continue as Israel explores new ways to provide humanitarian aid to Hamas instead of through UNRWA.
Among the organizers of the protests is the Tikvah Forum, a group composed of families of Israeli hostages who favor a more hardline response to Hamas.
The IDF released a statement on the incident, stressing that physical proximity to the border area is "dangerous and forbidden."
If Israel halts aid to Gaza, nations mediating between Israel and Hamas would increase pressure on the terrorist organization, former National Security Council chief Giora Eiland argued.
'As an Israeli, when I tell the story of Israel, it’s not out of hate for Palestinians but rather love for my country'.
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.
Israel has closed its commercial and pedestrian crossings into Gaza, forcing humanitarian assistance to travel into the enclave through Egypt’s Rafah crossing.
Until now, limited quantities of aid have been delivered from Egypt through the Rafah crossing, which is ill-equipped to process large numbers of trucks.