The 26-year-old driver, who gave himself up to police after Saturday's tragedy, is not yet in a condition to be questioned, the prosecution said.
“The wounds will be difficult to heal, it will take time,” Interior Minister Johanna Mikl-Leitner said during a visit on Sunday to the scene on one of Graz's main shopping streets.
“What happened here is unthinkable. There is no excuse for it.”
Shoppers described scenes of terror as the car ploughed into the pedestrian street at over 100 kilometres (60 miles) an hour, forcing people to scramble out of harm's way.
During his rampage, the driver briefly got out of the car wielding a knife and wounded two people, the head of the investigation Kurt Kemeter said.
A four-year-old boy, a 24-year-old woman and a man, 28, were killed and another 34 were injured. Six remain in hospital, and three of them are reported to be in a critical condition.
The suspect, a father of two, was described as a “mentally unbalanced” man, according to governor Hermann Schützenhöfer.
Regional police chief Josef Klamminger ruled out a terrorist motive.
“We can say clearly and without doubt that it is an isolated incident without political or extremist motivation,” he said.
An initial investigation revealed that the driver showed symptoms of “psychosis” and was known for “acts of violence”, which led to him being barred from the family home at the end of May.
Austrian President Heinz Fischer said he was “deeply shocked” by the attack.
“People were screaming in panic and rushed into the stores to seek shelter,” said one woman at the scene, adding that the crashing of metal chairs on cafe terraces sounded like “a shootout”.
Several thousand people attended a memorial on Saturday night in Graz, where a huge sign reading “Graz in mourning” was erected on the main square.
Flowers, candles and soft toys were laid in memory of the three who died.