Pillay said in a statement: "1.5 million Palestinian men, women and children have been forcibly deprived of their most basic human rights for months."
Sherine Tadros, Al Jazeera's correspondent in Gaza, said the latest closures have blocked the delivery of about 17 truckloads of aid to UN Relief and Works Agency (Unrwa) distribution centres in the Gaza Strip.
"They [Unrwa] says that the 11 truckloads they received on Monday will be enough to continue distribution to the 750,000 Palestinians in Gaza for about a week," Tadros said.
"After that, the UN says it simply does not know what it can do."
Oxfam International, a British humanitarian agency, said in a statement that "only the bare minimum of goods have entered Gaza in the past couple of days".
The organisation "fears a serious worsening once again of the humanitarian situation if urgent action is not taken", Barbara Stocking, Oxfam's chief executive, said on Tuesday.
Before allowing some aid trucks into Gaza on Monday, Israel had not allowed relief workers to deliver aid to Gaza since November 4.
In the past few days, Israeli has also cut off fuel supplies into Gaza, forcing the closure of the territory's main power plant.
"The power cuts are affecting Palestinians and Palestinian businesses. The water sewage treatment facility is also affected. Sewage is seeping into Palestinian refugee camps and the sea around Gaza."
The Israeli-imposed restrictions follow a series of Israeli raids into Gaza, in which more than a dozen Palestinian fighters have been killed.
Several Israelis have been injured by rockets fired by Palestinian fighters into Israel in recent days.
Hamas fighters fired mortar bombs at Israeli soldiers who were searching for explosives near the Gaza border on Tuesday, Israeli military and Hamas said.
Also on Tuesday, Israeli tanks entered about a quarter of a mile into the Gaza Strip, east of the city of Rafah, residents and Gaza security officials said.
On Monday, Mahmoud Abbas, the Palestinian president and leader of Fatah, called on Israel to stick to a five-month-old Egyptian-brokered truce with Hamas, which has de facto control of Gaza.
The truce between Israel and Hamas, which is not involved within the US-sponsored peace talks, is due to expire next month.