Italian Prime Minister Silvio Berlusconi has said the car carrying an Italian agent killed by US fire had stopped as soon as it was signalled.
Mr Berlusconi says the US knew of Mr Calipari's journey to the airport
His statement contradicts US accounts of the incident in Iraq in which Nicola Calipari was shot taking freed hostage Giuliana Sgrena to Baghdad airport.
Mr Berlusconi said the US must accept responsibility to restore relations.
He said his government had demanded "maximum co-operation" from the US, and a joint inquiry had been promised.
The US military in Iraq has begun an investigation led by Brig Gen Peter Vangjel into Friday's shooting. It is expected to take up to four weeks to complete.
US military: Car approaches checkpoint at high speed
Troops attempt to tell driver to stop with arm signals, lights and warning shots
Soldiers shoot into engine
Italian government: Italy makes all necessary contacts with the US for safe passage
The driver stops immediately when a light flashes 10m away
At the same time, shots are fired for 10-15 secs into the car
Relations between the US and Italy have been strained by the incident.
Speaking to the Italian Senate, Mr Berlusconi said: "Only a frank and reciprocal recognition of eventual responsibility is the condition for closure of the incident, which was so irrational and caused so much sorrow."
The prime minister said the US military had authorised the Italian journey to the airport.
An agent travelling in the car with Mr Calipari had given an account of events which conflicted with the version given by the US military, he added.
"A light was flashed at the vehicle from 10m away," Mr Berlusconi said. "The driver at this point stopped the car immediately and at the same time there was gunfire for about 10 or 15 seconds.
"A few shots reached the vehicle and another one reached and killed Mr Calipari," he said.
"This reconstruction of events has been made according to what has been witnessed by another agent who was with Mr Calipari and does not coincide totally with what has been communicated so far by the US authorities."
The US says the vehicle carrying Ms Sgrena to the airport was "travelling at high speeds" and "refused to stop at a checkpoint". Soldiers fired at the engine when the driver failed to stop after several warnings, military officials said.
Nicola Calipari died protecting freed journalist Giuliana Sgrena
Mr Berlusconi said he had spoken to US President George W Bush, who had promised to co-operate in finding out who was responsible.
He said the idea that Mr Calipari had been killed by friendly fire was "painful" but added he was certain the US "has no intention of evading the truth".
"I'm sure that in a very short time every aspect of this will be clarified," he said.
"Our friendship with the US is strong and loyal, and we have the duty to demand from them the utmost truth."
Mr Calipari has become a national hero and Italy's leaders joined hundreds of fellow citizens at his funeral.