Faleh Almaleki allegedly accelerated his Jeep in a car park in Phoenix, Arizona, hitting Noor as well as her boyfriend's mother, before fleeing the scene - and the country.
The mother survived but Noor died after two weeks in a coma.
Accused: Faleh Hassan Almaleki, left, is charged with murdering his 20-year-old daughter Noor, after she died when he ran over her in his Jeep in October 2009. It is alleged he was angry that she was 'too Westernised'
The 50-year-old faces life behind bars if he is convicted of murder.
Prosecutor Laura Reckart told the 16-strong jury that Almaleki was increasingly incensed at his daughter's failure to obey him.
In her opening statement, she told jurors that he believed she had dishonoured the family by becoming too Westernised.
And, when he saw her by chance while visiting a state Department of Economic security office in Peoria on October 20, 2009, his rage overflowed, she has claimed.
Almaleki left the office, got in his Jeep and waited for his daughter to emerge as he seethed, Mrs Reckart said.
'He revved and raced that car right out of the parking spot in a premeditated act and ran them over,' she continued.
However, according to his defence team, led by Elizabeth Mullins, what happened was accidental.
As he accelerated to about 20 mph (32 kph), a startled Amal Khalaf - the mother of Miss Almaleki's boyfriend - jumped in front of the Jeep.
Almaleki swerved but couldn't avoid hitting the woman, who thankfully survived.
But the Jeep then jumped a curb and ran over his daughter, causing severe injuries that would kill her 13 days later.
Almaleki's lawyer claims it was an accident and that although he should be held responsible, he is not guilty of premeditated murder.
The case caused outrage after prosecutors deemed Miss Almaleki's death an 'honour killing'.
Almaleki is from a small Southern Iraqi town near Basra.
He and his family were relocated by the U.S. military to Saudi Arabia and then the U.S. in the mid-1990s.
Almaleki later got a job as a truck driver in the Phoenix area and his wife worked for the military.
He wanted his daughter to adhere to Iraqi traditions, but she wanted to be a typical American girl, the court heard.
When she was 17, she refused to enter an arranged marriage in Iraq - a move that enraged her father - the court records showed.
She moved into her own apartment aged 19 and began working at a fast-food restaurant but quit after her parents kept showing up at her workplace, insisting she return home.
Later in 2009, she moved in with her boyfriend and his parents, Reikan and Amal Khalaf, after saying her parents had hit her.
'The relationship between Noor and her father had become very strained,' Mrs Reckart, for the prosecution, said.
'Noor wanted to live her life like those her own age, but the defendant would not allow it.'
That may be true, Mrs Mullins said from the defence bench, but Mr Almaleki would never intentionally harm his first born.
'For a real (Iraqi) man, for a real father, having that child is the most important thing in his world,' she told jurors.
But the prosecutor said Almaleki's actions after he fled the scene - which included travelling to Mexico and then on to London, where he was caught - show he was running away and knew what he had done was wrong.
Mrs Mullins said he simply panicked and left at the urging of his family.
When he was caught, he tried to explain to police that what had happened was an accident, telling them, 'If my daughter dies, I kill myself', she said.
When Miss Almaleki eventually died, her father was marked for suicide watch in the jail, she added.
Prosecutors plan to begin calling witnesses in the case today and the trial could take several weeks.
Almaleki has pleaded not guilty to first-degree murder, attempted first-degree murder, aggravated assault and leaving the scene of a serious injury accident.