"The second demand is to bring the proper justice and the proper punishment to those members of the Blackwater company that have committed unjustifiable crimes against innocent Iraqi citizens," he said.
"I also would like to relay the justifiable demand of the Iraqi Islamic resistance for the complete withdrawal from Iraq, so Iraq can become a sovereign nation."
Asaib Ahl al-Haq was believed to be responsible for the kidnapping of Peter Moore, a British IT expert, and his four bodyguards in May 2007.
Moore was released last year, but three of the bodyguards have been confirmed killed and the fourth is still missing.
A senior leader of Asaib Ahl al-Haq was reported as telling the Reuters news agency on Saturday that the abducted contractor, whom he did not name, was seized because the government was not keeping a promise to free the group's supporters from prison.
Such a deal was widely believed to have been behind the release of Moore in December, despite Iraqi and British government denials of a link.
"The government is carrying out many violations against us," the leader was quoted as telling Reuters.
If the kidnapping of Salomi confirmed, it will be the first of a foreigner in Iraq since the group took Moore.
The US military has not commented on the video.