The government, however, has rejected the demand.
Sergeant Steve Roberts, 33, had died after being shot in the chest in Basra in southern Iraq on 23 March last year.
Samantha Roberts, his widow, on Thursday released an audio diary made by Roberts in which he complained about the poor level of supplies available to British soldiers.
He said in the diary that lack of adequate equipment was "disgraceful".
"We know we are going to have to go to war without the correct equipment," he said just before the US-led invasion of Iraq.
Ministry admits lapses
The British ministry of defence has admitted in the past that there had been problems in ensuring front line forces received equipment supplies.
"Clearly the report that came out, to me suggests that they are responsible... for him not having a flak jacket being originally issued, and then withdrawn because there were not enough to go around"
The opposition Conservative Party said body armour would have saved Roberts, while his widow believes that the ministry is responsible for his death.
"Clearly the report that came out, to me suggests that they are responsible... for him not having a flak jacket being originally issued, and then withdrawn because there were not enough to go around," she told BBC television.
Conservative leader Michael Howard said Roberts' death amounted to a "grave dereliction of duty" and called on Hoon to quit.
But a junior defence minister, Ivor Caplin, said the matter was not an issue to resign over.
"I don't think that when we have had the odd glitch or shortcoming in our processes... that is necessarily a matter for ministerial responsibility," he told the BBC.
A report from the National Audit Office in December had found that British troops were sent ill-equipped to Iraq.
The office said that while the logistics effort had been successful overall, the means of tracking supplies had been ineffective.