In a statement, Ehud Olmert said he "will not yield to the extortion of the Palestinian Authority and the Hamas government, which are led by murderous terrorist organisations".
Olmert also dismissed the possibility of negotiating with those holding Corporal Gilad Shalit.
Earlier on Monday, the Palestinian groups that abducted the soldier gave Israel until 0300 GMT on Tuesday to meet their demands, threatening unspecified consequences if it refused.
In previous statements, the groups demanded that Israel, as a first stage, release Palestinian women and youths in its prisons in exchange for information about Shalit, seized in a cross-border raid on June 25.
The Israeli defence minister, Amir Peretz, said that the Hamas office in the Syrian capital Damascus, led by exiled leader Khaled Meshaal, "is the main address that bears responsibility" for the abduction.
"I suggest to [Syrian President] Bashar Assad, who is trying to turn a blind eye, that he open his eyes, as the responsibility is at his doorstep," Peretz said.
Meanwhile Assad renewed his support for the Palestinians during a speech to Syria's highest political body, the National Progressive Front.
Assad reportedly dismissed
"Israel's aggressive stand and its unjust accusations against [Palestinian] national forces serve to increase our commitment to Arab rights," he said in his first reaction to the crisis.
A senior Syrian politician reportedly said that during the meeting Assad dismissed Meshaal's involvement in the soldier's capture and said a prisoner exchange was the best way to resolve the issue.
Last week, Israeli aircraft flew over Assad's summer palace in a warning over his hosting of Hamas.
Earlier on Monday, a small Israeli force entered the Gaza Strip for the first time since Shalit's capture.
Tanks and armoured vehicles crossed the northern Gaza border, along with a detachment of Israeli troops, to search for explosive devices and tunnels, militiary sources said on Monday.
Witnesses said about 25 Israeli tanks crossed into Gaza. However, Israeli military sources said the incursion was not a large-scale ground operation and would be limited in duration.
Also on Monday, an Israeli helicopter gunship fired a missile into a Hamas office in Gaza City, damaging a building the Israeli army said was being used by Palestinian fighters.
However, the Fatah movement of Mahmoud Abbas, the Palestinian president, said the office was a charity and near to a Hamas office. There were no reported casualties.
Fatah said the attacked building
was a charity
Witnesses also reported two missiles landing in open areas in northern Gaza, near the town of Bait Lahiya.
Monday's strike came hours after Olmert ordered the military to intensify its operation to free a captured Israeli soldier.
About 5,000 Israeli troops remain poised on the Gaza border in the largest Israeli military operation since it pulled it troops out of the tiny coastal territory last year.