Footage broadcast by Aljazeera on Tuesday showed the captive with three hooded men, and documents identifying him as Turki Khalifa al-Braizat.
The hitherto unknown group called Brigades of United Lions said it was "giving the firm employing the Jordanian driver, who admitted to ferrying fuel to the US army, 48 hours to announce it is ceasing operations in Iraq, otherwise they would kill the hostage".
Dozens of foreigners, many of them truck drivers, have been seized in Iraq by groups seeking to put pressure on their employers or governments, to leave the war-torn country.
Three Jordanians taken captive along with a Sudanese national were released by their captors on 6 September.
Jordan had also secured the release of eight of its citizens captured in Iraq in July and August through mediation efforts and in some cases a ransom was paid by relatives of the captives in exchange for their freedom.
Some of the captive foreigners have been killed by their captors.