In a statement received by AFP in Gaza City, the armed group, loosely affiliated to the Fatah movement, called for "the end of the Israeli offensive" and "the lifting of the blockade" in the territories.
"If our demands are not met, the Al-Aqsa Brigades will kill the kidnapped soldier," the statement said, without giving any details.
The soldier was seized in the West Bank, the group said. They named him as Hoffmann Kfeir Samuel, 24, a resident of Hulul, south of Tel Aviv.
There was no immediate Israeli response.
The group also demanded the release of Hamas ministers and deputies arrested by Israeli forces following the earlier kidnapping of 19-year-old army corporal Gilad Shalit, along with the release of "all the other detained Palestinians".
The Israeli army launched a major air and ground offensive earlier this week, largely in the Gaza Strip, following the capture of Shalit.
Meanwhile, three Palestinian groups that had earlier captured Gilad Shalit have demanded in a statement on Saturday that Israel free 1,000 Arab prisoners held in its jails and end an offensive in Gaza.
The document titled: Military Statement Number Two, which was faxed to news agencies, did not specify whether the demands were for the release of Corporal Gilad Shalit.
Palestinians with pictures of
their relatives in Israeli jails
But a spokesman for the military wing of the governing Hamas group, one of the three factions, said they were conditions for freeing Shalit.
Israel has said it will not discuss releasing prisoners in exchange for Shalit.
It was the second statement released by the groups since Shalit was abducted in a cross-border raid on Sunday.
"We are declaring to the public our just and humanitarian demands," it said.
It repeated an earlier demand for the release of women prisoners and minors in exchange for information on Shalit, but later made the added request for Israel to free 1,000 "Palestinian, Arab and Muslim prisoners".
It said these would have to include all Palestinian faction leaders as well as humanitarian cases.
Doubts over release
The statement cast doubt on hopes voiced by mediators that Shalit could be freed soon.
"In spite of the good efforts of the mediators who tried in silence to speed up the treatment of this humanitarian matter, the enemy and their political leadership are still under the pressure of the security and military command," it said.
"The escalation and arrogance mean the enemy will be responsible for the bad consequences," it said. There was no specific threat to Shalit.
"The escalation and arrogance mean the enemy will be responsible for the bad consequences"
The statement by the armed groups who kidnapped Shalit
Besides Hamas's armed wing, the Popular Resistance Committees and Army of Islam, too, participated in the kidnapping of the soldier. They have not said their statements are an admission that they are holding him.
Israeli tanks entered the southern Gaza Strip this week in the biggest push into the territory since Israel pulled out troops and settlers last year after 38 years of occupation. The offensive, which has included air strikes and artillery, is intended to force Shalit's release.