The military initially described Ahmad Kahla, who was 45, as a terrorist who had tried to carry out an attack.
However, Israeli media said a leaked preliminary report found the father of four was killed unnecessarily.
A video showed a scuffle with soldiers before Mr Kahla was fatally shot.
"How can I explain to you what I feel psychologically or physically?" his son, Qusai, who is 20, told the BBC.
"We were on our way to work together before this happened with the soldiers. Then we were separated, and I only saw my father again when they brought his body to the hospital."
Witnesses said a line of traffic had formed as Israeli forces subjected them to lengthy vehicle checks to enter or leave the village of Silwad, north of Ramallah.
Qusai Kahla recounted how his father pulled over when he saw that a driver on the other side of the road had a flat tyre and got out to help him.
About 10 minutes later, he said, when he rejoined the queue, angry Palestinian drivers honked their horns thinking he was pushing in. Then a soldier came towards them and threw a sound bomb at the car's front windscreen.
"My father opened the window and started arguing with the soldier. He asked him: 'Why did you hit the car?' As one soldier went to my father's side, another came to me," Qusai said.
"He had pepper spray and he sprayed it on us. Then he opened the door and took me out of the car. They took me away and I could hear fighting and a gunshot. I couldn't see anything. Then after 10 minutes, I heard an ambulance."
Israel's public broadcaster, Kan, quoted an initial military inquiry as saying that a tussle ensued as a soldier tried to drag Mr Kahla from his vehicle.
He apparently tried to pull at a gun which had been used to hit him and was then shot.
On 15 January, the Israeli military first said that a Palestinian man had approached soldiers with a knife near Silwad.
It later updated its version of events to say that soldiers had ordered a man to stop in his vehicle and used tear gas when he refused. It said that there had then been a confrontation, and that he had tried to snatch a weapon from a soldier before he was shot.
Kan said that the Israeli brigade commander overseeing the initial investigation had concluded Mr Kahla was not planning an attack. It quoted the commander as saying: "The incident was not supposed to end with a man dead."
Asked for a response, the Israeli military said that a military police investigation had been opened into what happened and that its findings would be transferred to the Military Advocate General's office.
It is unusual for military prosecutors to press charges against troops in such cases.
The Palestinian Authority's foreign ministry strongly condemned what happened as a "heinous crime of execution".
In total, 18 Palestinians have been killed by Israeli forces in the first three weeks of this year, including militants and civilians.