Private Jacob Kovco, who died in a Baghdad hospital a week ago becoming Australia's first military fatality in Iraq, was due to be buried with full military honours near Melbourne.
However another body, possibly that of a Bosnian soldier, was flown to Australia while Kovco's body remained in a mortuary in Kuwait.
Howard was woken in the middle of the night on Wednesday to take a phone call from Kovco's widow, Shelley.
The prime minister said Shelley Kovco was very angry about the mix-up and demanded every effort be made to return her husband's body as soon as possible.
Howard said during a radio interview: "I just want to say how incredibly sorry I am for what has happened and I wish in some way I could have altered it, but I can't."
the government, severely embarrassed by the mix-up, plans to send a senior defence official and civilian pathologist to Iraq next week to investigate.
The defence minister, Brendan Nelson, said a private mortuary in Kuwait appeared to be responsible for the mistake, and that the Australian government was investigating how Kovco's coffin was swapped for another.
The Sydney Morning Herald newspaper reported that
Howard later said he felt extremely sorry for Kovco's mother, and that she had declined his offer to fly to Kuwait on his private jet and escort her son's remains home later this week.
"John Howard, you're nothing but a shocker"
Kevin Backman, victim's cousin
"She was very distressed. She was very angry, she was very upset, and all of that is totally understandable," Howard told Southern Cross Broadcasting.
Kovco's cousin, Adam Backman, called the mix-up an "absolute disgrace".
"It's shocking. It's a disgrace. The government and the Australian army should be ashamed of themselves," he told local radio. "John Howard, you're nothing but a shocker."
Earlier in the day, Nelson denied reports that the 25-year-old soldier had shot himself while cleaning his gun.
"He wasn't in fact cleaning his weapon," Nelson said.
He was unable to explain how the accident occurred, saying only that Kovco was in his room with two other soldiers and his Browning pistol at the time of the incident.
Australia has more than 900 troops serving with the US-led forces in Iraq.