UNRWA official Natalie Boucly celebrated the arrest warrants against Israeli Prime Minister Benjamin Netanyahu and former Defense Minister Gallant as the start of "accountability."
The blockade and tightly regulated aid have led to the resurgence of the smuggling industry.
Bernie Sanders and other progressive senators call to block US offensive weapon sales to Israel, citing the Gaza humanitarian crisis.
In speaking with reporters prior to a monthly meeting of foreign ministers, Borrell said, that it was important to pressure Israel.
This is in response to rising civilian anger at price gouging and aid seizures.
The Israeli military "intends to continue its tireless efforts to improve the humanitarian situation in the Strip during the ongoing conflict," a statement said.
Washington warned Israel last month that military aid could be limited until progress was made. It did so based on Memorandum 20, which links such aid to humanitarian actions.