The ultimatum came as Israel made good on its promise to continue its military offensive until the soldier was freed, firing artillery shells and missiles into the coastal strip and massing troops and tanks along the Gaza-Israel border.
The militant groups, in a statement posted on the Web site of the ruling Hamas party's military wing, did not expressly say what the consequences would be, but implied the soldier could be killed. Israeli government spokesman Asaf Shariv said officials were studying the statement.
"We give the Zionist enemy until 6:00 tomorrow morning, Tuesday, July 4, (11 p.m. EDT Monday)" the groups said in their statement, which was also faxed to news agencies.
"If the enemy does not respond to our humanitarian demands mentioned in previous leaflets on the conditions for dealing with the case of the missing soldier ... we will consider the soldier's case to be closed," it said. "And then the enemy must bear all the consequences of the future results."
The Palestinians apparently face a deadline of their own, reports CBS News correspondent Sheila MacVicar.
"Egyptian mediators, who have been key to the whole attempt to free the soldier, have told the Hamas leadership outside of Gaza that they have until this evening to respond, or all talks are off," MacVicar said.
Cpl. Gilad Shalit, 19, was captured June 25 in a cross-border raid by the military wing of the Palestinians' ruling Hamas party, and two allied groups, the Popular Resistance Committees and the Army of Islam.
Prime Minister Ehud Olmert has said repeatedly that Israel would not negotiate Shalit's release. But Israel has swapped prisoners in the past to win the release of captured citizens, alive and dead, and privately, political and defense leaders have not ruled out releasing prisoners who weren't involved in attacks on Israel, officials said on condition of anonymity because of the sensitive nature of the discussions.
In the week since the Israeli soldier was captured, Palestinians have come together and have come behind their government, sharing the view that something must be gained for all this pain, MacVicar reports.
At the very least, Palestinians are hoping the captive soldier will be traded for the release of hundreds of Palestinian prisoners in Israeli jails. That is a demand that is almost universally popular, MacVicar reports.
In the meantime, Israel has sent tanks, troops and warplanes to attack Gaza over the past week in an effort to press militants to free Shalit. Intensive efforts to mediate his release, involving Egypt and other countries, so far have not been successful.
There has been no sign of life from the soldier since his seizure, and no concrete evidence of his condition, though Israeli officials have said they think he is alive. The Shalit family had no immediate comment on the ultimatum.
The pan-Arab newspaper Al Hayat reported Monday that an Egyptian security team had visited the kidnapped Israeli soldier in Gaza and said he was being treated by a Palestinian doctor for "three bullet wounds." The paper did not say when the visit took place.
Shalit's captors initially demanded the release of about 500 women and children prisoners held in Israeli jails. They later raised their demands to include an additional 1,000 prisoners. Israel is currently holding about 9,000 Palestinians.
The ultimatum requires Israel only to "start" freeing the prisoners by Tuesday morning.
Hamas, which has killed hundreds of Israelis in suicide bombings, has refused to renounce violence or recognize Israel since taking power in March. But the Hamas government and Hamas leaders based in Syria have denied responsibility for the soldier's capture.
A spokesman for the Hamas government said the ultimatum was "a message to Israel that all its military escalation will not get it anywhere."
"If it continues every day to kill and target and attack, it won't get the soldier, alive or dead," spokesman Ghazi Hamad said. "That is the meaning of the message."
In their statement, Shalit's captors accused Israel of not "learning lessons" from the cases of other kidnapped soldiers. The last Israeli soldier kidnapped by Hamas, Nachshon Wachsman, died in 1994 in an Israeli commando raid on his captors' Jerusalem hide-out.
In an op-ed piece in the Haaretz newspaper, Wachsman's mother, Esther, wrote how the family, through Shalit's capture, was "once again going back" to those dark days when her son was being held. She criticized Israel's leaders for a lack of candor in dealing with abduction cases.
"I am not calling for the release of murderers, but they (Israel's leaders) should not insult our intelligence because they have negotiated and they have given in to terror," Esther Wachsman wrote, referring to cases in which Israel has swapped prisoners in the past.
When it launched its first large-scale military action in Gaza since withdrawing from the strip last summer, Israel said it declared purpose was to lean on militants to release Shalit. In statements since, government officials have said the campaign is also meant to topple the Hamas government and stop gunmen from launching rockets at southern Israel.
Early Monday, Israel massed tanks and troops across from northern Gaza, and pounded the area with artillery. At daybreak, a small force of Israeli tanks entered northern Gaza, but the military said it was a "limited" mission to find explosives and tunnels near the border fence.
Additional Israeli troops moved into place across from northern Gaza on Sunday, showing clear preparations for an invasion that was put off last week to give diplomacy more time. For months, Palestinian militants have launched homemade rockets at Israeli villages near the border fence, and Israel has been unable to stop the barrages with repeated airstrikes and artillery attacks.
Also Monday, Israeli troops killed one gunman after he and another militant approached soldiers in northern Gaza, the military said, without providing further details. Palestinian medical officials confirmed that an armed Palestinian man had been killed in Beit Hanoun.
Separately, Israeli artillery hit a house on the outskirts of Beit Hanoun, Palestinians said, slightly injuring one person. Around the same time, Israeli aircraft hit several targets around the Gaza Strip, including a building in Gaza City where the Al Aqsa Martyrs' Brigades, a violent offshoot of Palestinian President Mahmoud Abbas' Fatah movement, has an office, officials said.
A missile struck the second floor of the building, setting it on fire. No one was in the office at the time of the attack. A family living on the first floor escaped.