The regiment made its call after Armed Forces minister Adam Ingram yesterday told the House of Commons that the pictures were categorically not taken in Iraq.
At a press conference in Preston today, the Regiment said that the Daily Mirror must apologise for publishing them and endangering British troops.
Colonel David Black said: "It is time the ego of one editor is measured against the life of a soldier."
Col Black said: "The regiment is made up of young men and women from this county. They went to Basra last year and faced incredible danger in incredible difficulties. It's they who have been besmirched, the good lads and good lasses of the regiment. It's they who should have the apology."
He added: "It's up to the readership, the board of directors and the shareholders to put pressure on to get an apology."
Colonel Black, a former commander of the Queen's Lancashire's, said that the Daily Mirror pictures were a "recruiting poster" for Al-Qaeda and had put the lives of British soldiers at stake.
The press conference followed news that one of the soldiers who spoke to the Daily Mirror, dubbed Soldier C, is to appear on Sir Trevor McDonald's programme, 'Tonight with Trevor McDonald', tonight.
During the press conference, the Army demonstrated the aspects of uniform and equipment which it said proved the photographs were fake.
Reporters were told that the Bedford truck used in the photographs was located in a Territorial Army base in Lancashire and had never been in Iraq.
The regiment's Brigadier Geoff Sheldon, speaking alongside Colonel Black, said that the regiment's reputation had been damaged and said the paper should apologise for the "terrible" damage it has done.
"It wasn't a British soldier degrading an Iraqi. It was a mocked-up fake, not even taken in Iraq."
The Special Investigations Branch of the Royal Military Police is attempting to track down the soldiers A and B who sold the hoax pictures to the paper.
The media world is divided on whether Morgan should go. Andrew Gowers, editor of the Financial Times, said yesterday that in Morgan's place he would go.
Former Mirror editor Roy Greenslade earlier today said he thought the Mirror had done all that was necessary. "Piers Morgan can defend himself by saying he's performed a public function," he told Radio 4's 'Today' programme.
Former Sunday Times editor Andrew Neil, however, was unequivocal in his condemnation. He called for Trinity Mirror to sack Morgan.
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