The attack on the armoured personnel carrier led to gruesome scenes, as militants paraded in the streets with what they said were body parts belonging to the dead soldiers.
Israeli Prime Minister Ariel Sharon chaired a meeting of the security cabinet Tuesday night to weigh a response, but no decisions were taken, public television reported.
"We hold body parts belonging to your soldiers that were scattered in the streets of Gaza and we are setting conditions to return them to the usurper," said a joint statement from the Al-Aqsa Martyrs Brigades, an offshoot of the mainstream Fatah group, and the Al-Quds Brigades, the armed wing of Islamic Jihad.
They did not specify what their conditions were.
Israeli government spokesman Avi Pazner described the militants as "barbarians."
"Parading with what is presented as the desecrated remains of our soldiers is evidence of a bestiality worthy of inhumane barbarians," he told AFP.
The attack was claimed by the leading Palestinian Islamist movement, Hamas, dealing the military one of its heaviest losses in months.
Sharon said earlier: "We are paying a heavy price to ensure the security of Israelis. We are fighting a cruel enemy and we will relentlessly fight him wherever he operates and wherever he hides."
In Washington, State Department spokesman Richard Boucher said "we're all very concerned, shocked by the reports we've seen out of Gaza."
He said Secretary of State Colin Powell had called his Israeli counterpart, Silvan Shalom, to discuss the situation.
In New York, the spokesman for UN Secretary General Kofi Annan said he was "deeply disturbed by the alarming violence in Gaza City" and "urges both sides to desist from further violence."
Fred Eckhard said Annan "is particularly concerned by the heavy fighting in densely populated neighbourhoods, which is exposing more Palestinian civilians to death and injury."
He also urged both parties to "heed the calls of the quartet and the international community to resume negotiations to end the conflict."
The "quartet" comprises the United Nations, the United States, Russia and the European Union.
The violence was some of the worst seen in Gaza Strip in several months.
Palestinian militants detonated explosive charges against an armoured personnel carrier after forcing it to a halt by firing an anti-tank rocket during a raid in the Zeitun neighbourhood.
The Israeli army announced all six crew members were killed in the blast.
"Six IDF soldiers were killed this morning during an IDF (army) operation to target Qassam workshops in Gaza City, when an armoured personnel carrier was struck by an explosive device planted by Palestinian terrorists," it said.
Israeli troops moved through the narrow streets of Gaza City in a "search and rescue" operation to retrieve the bodies of their dead comrades.
"We attach great significance to the retrieval of the bodies, which we want to be buried in a dignified way," a senior military source told AFP.
The Israeli chief of staff, General Moshe Yaalon, said troops were "conducting an operation in Gaza to get the bodies or the remains of the soldiers which are still there.
"We will not in any way tolerate those who have profaned the bodies of our soldiers, and we will not engage in negotiations to recover their remains," said the general.
Troops, tanks and aircraft were involved in the operation, which would continue "as long as is necessary".
The Israeli authorities asked for the help of the International Committee of the Red Cross in retrieving the bodies, an official from Sharon's office said, while the Palestinian Authority called for those holding the remains to return them to Israel.
Gun-battles that broke out overnight left eight Palestinians dead in Zeitun, including two members of the Hamas movement, Palestinian medical sources said.
A 12-year-old child was also killed when an Israeli helicopter fired a missile on a car near a market in the Gaza City district of Shajayah, the sources added.
Israeli military sources said the strike targeted a group of gunmen about to attack Israeli soldiers.
The Gaza flare-up came amid speculation over the fate of Sharon's plan to pull out of the flash-point Gaza Strip.
The "disengagement" plan includes a troop withdrawal from most of Gaza and the removal of all 21 settlements there, but the unilateral measure was rejected by Sharon's own Likud party on May 2.
An opinion poll released Tuesday showed once again that a majority of the Israeli public would support a Gaza withdrawal, which has also received qualified international backing.