The video, aired on the Al-Arabiya satellite channel, showed Mahmud Sulayman Saidat, a driver for the Jordanian embassy in Baghdad who was kidnapped on Tuesday by gunmen.
Saidat read a statement while sitting on the floor, surrounded by three armed men.
"I ask the Jordanian government to withdraw its diplomatic mission from Iraq and not to co-operate with this illegitimate government [in Baghdad]," he said.
He also called for Jordan to release Sajida Mubarak al-Rishawi, a would-be suicide bomber. Her explosives belt failed to go off at attacks on hotels in Amman on 9 November in which 60 people died.
A banner hanging in the background identified Saidat's kidnappers as a previously unknown group, the Hawk Brigades.
Al-Arabiya said the statement in the video gave the Jordanain government three days to meet the demand. It did not specify whether the militants threatened to kill Saidat if the deadline were not met.
Saidat, a Jordanian citizen, was kidnapped by more than a dozen masked men as he drove to work at the embassy in Baghdad.