Local people demonstrated against the deaths in Kut
The US military is "deeply saddened" by the outcome of its raid in southern Iraq in which two people were killed, a senior US officer has said.
Colonel Richard Francey, expressed his condolences for "a terrible tragedy" that took place in Kut on Sunday.
Iraqi Prime Minister Nouri Maliki denounced the operation as a crime and a violation of the security pact governing the US presence in Iraq.
The US says the raid was approved by Iraqi officials, as the pact requires.
Iraq has detained two of its army commanders who allegedly authorised the operation without the knowledge of officials.
A policeman and a woman were shot dead and six people detained.
On Monday the Americans freed their detainees.
Among them was Muamar Ahmed Abdul-Munim who said his wife and his brother had been killed. "We want to prosecute the soldiers who killed our loved ones," he said.
Mr Abdul-Munin says that he and the other detainees were hooded and taken by helicopter to an unknown destination.
A female US officer then questioned them over their alleged links to Iran and Iranian-backed militia.
The BBC's Jim Muir in Baghdad says it is the most serious dispute between the US and Iraq since the agreement came into force at the start of the year.
US forces stormed buildings in Wasit province early on Sunday morning.
The US military had said the raid, against a weapons smuggler and "network financier", had been "fully coordinated and approved by the Iraqi government".
They said soldiers had shot and killed "an individual with a weapon" outside the house and that the woman who died had "moved into the line of fire".
In a statement read on state TV, Mr Maliki had called on the US to "release the detainees and hand over those responsible for this crime to the courts".
The incident caused uproar in Wasit, where provincial governor Latif Hamad al-Turfah echoed Mr Maliki's condemnation.
He said local government and officials had been "surprised that these forces carried out the raid in breach of the agreement signed between the Iraqi and US governments".
The chairman of the provincial council, Mahmud Abd al-Rida, said the raid had embodied the "meaning of the occupation".
"Their claim of friendship and early withdrawal from our dear land, according to the security agreement signed by the two Iraqi and US parties, is meaningless," he said.
The complicated Status of Forces Agreement was signed in November last year and came into force in early 2009.
It requires all military operations in Iraq to have the government's approval and allows for US soldiers to face trial if they commit crimes off base.
The US currently has more than 140,000 troops in Iraq, and combat troops are due to pull out of Iraq's cities by the end of June.