The vote gives Israel and the Palestinians an additional five months to undertake their investigations, the two sides having been given an initial three months following the previous vote in November.
The latest resolution asks Ban Ki-moon, the UN secretary general, to report back to the assembly "within a period of five months on the implementation of the present resolution, with a view to the consideration of further action, if necessary, by the relevant UN organs and bodies, including the Security Council".
Those voting against the resolution included Israel, Canada, Nauru, Panama, Micronesia, Macedonia and the United States.
No EU member voted against the text, with some voting yes and others abstaining.
In the wake of the 22-day Israeli onslaught on Gaza, a UN panel accused mainly Israel, but also Hamas, the de facto rulers of the Palestinian territory, of war crimes and possible crimes against humanity.
About 1,400 Palestinians and 13 Israelis were killed during the Israeli military attack.
A key finding of the UN report, headed by Richard Goldstone, a South African judge, was that Israel used disproportionate force in response to rocket attacks by Gaza-based fighters and failed to take adequate measures to protect civilians during its onslaught.
In the report, published in September last year, Goldstone recommended that both sides face possible prosecution before the International Criminal Court in The Hague if they failed to conduct credible, independent investigations within six months.
Cath Turner, Al Jazeera's correspondent at the UN, said that the text of Friday's resolution had not contained the word Goldstone within it.
She said: "Australia ... made mention of this during its address, saying that it appreciated the efforts made by the Arab countries to modify the language, which included the deletion of the word Goldstone.
Turner said that any mention of the Goldstone report had now become "politically loaded".
"[The report] has so much connotations of conflict and dispute, it has been seen, essentially, as a report damning Israel for its actions on the war on Gaza.
"So the removal of Goldstone did affect the vote for some of those countries and it was noted," she said.
Riyad Mansour, the Palestinian observer to the UN, expressed hope that the Palestinian side will be able to submit "a substantive report" within the next five months.
"If the Israelis continue to refuse an independent probe, they will then be isolated and this may pave the way for pressure on the Security Council to act," he said.
Israel, in a recent report on the allegations, denied violating international law, but admitted "tragic results" due to the "complexity and scale" of conducting a military operation in a heavily populated area.
It also said that two Israeli senior officers, a brigadier general and a colonel, had been disciplined for the firing of white phosphorus shells towards a UN compound.
A similar report from the Palestinian side said a commission of five well-known judges and legal experts had been set up to look into allegations of war crimes on its side during the conflict.