Israel's military categorically denied it was behind the strike in Deir al-Balah.
But the new attacks on Gaza sparked outrage, with thousands of Israelis taking part in an anti-government rally in Jerusalem.
Demo leader Shikma Bressler said: "It's time to end this madness before we don’t have anyone to save, before we don't have a country left."
The protesters blocked the main entrance of the capital as they marched towards Prime Minister Benjamin Netanyahu's residence.
They are angered both by Israel's renewal of hostilities with Hamas in Gaza and the government's continued efforts to oust Ronen Bar, chief of the Israeli security agency, Shin Bet.
The UN Office for Project Services said an "explosive ordnance was dropped or fired" at their building, which was in an isolated location.
Executive director Jorge Moreira da Silva dismissed the possibility of the attack being an accident and added: "UN personnel and its premises must be protected by all sides."
The Jerusalem rally came after Israel said it was resuming fighting in Ga...