Under "catastrophic" food insecurity, households suffer an extreme lack of food, leading to acute malnutrition in young children, an imminent risk of starvation, and deaths.
Israel has shut down both the Rafah and Kerem Shalom crossings for aid and people as part of its military operation in Rafah.
UNRWA chief Philippe Lazzarini said he had "intended to go into Rafah today, but was informed my entry had been declined.
"The start of Ramadan has been sad and covered in darkness, with the taste and stench of blood everywhere," said one displaced Palestinian man, Awni al-Kayyal, 50.
Pretoria said this may be "the last opportunity the Court shall have to save the Palestinian people in Gaza already dying of starvation, and now 'one step' from famine".
At least 15 children had died from malnutrition and dehydration at the Kamal Adwan hospital.
Israeli troops opened fire as Palestinian civilians scrambled for food supplies during a chaotic melee on Thursday that the Hamas-run territory's health ministry said killed more than 100 people in Gaza City.
The procession aimed to promote awareness among both Muslims and Christians, pushing them to unify and join efforts for a permanent truce.
A number of donor countries -- including Australia, Britain, Finland, Germany and Italy -- on Saturday followed the lead of the United States in suspending additional funding to UNRWA.
As the risk of famine grows, and more people are exposed to deadly disease outbreaks, a fundamental step change in the flow of humanitarian aid into Gaza is urgently needed, UN humanitarian agencies say.