The 26-year-old showed symptoms of “psychosis” in his first interrogation since he gave himself up to police after his Saturday rampage, though he remained “evasive” in his answers.
A spokesman for the prosecution, Christian Kroschl, said the man would be remanded in custody on Sunday and brought before a judge on Monday.
Three of the 34 injured were in a critical but stable condition a day after the incident that left Austria in a state of shock and mourning.
The suspect, a father of two, was described as a “mentally unbalanced” man, according to governor Hermann Schuetzenhoefer.
An initial investigation revealed that the driver was known for “acts of violence,” which led to him being barred from the family home at the end of May.
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.
Austrian President Heinz Fischer said Saturday he was “deeply shocked” by the attack.
“The wounds will be difficult to heal, it will take time,” Interior Minister Johanna Mikl-Leitner said during a visit 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.
“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.