Eleven victims remained in hospital in a stable condition on Tuesday, police said, adding that they all appeared to be recovering well.
British police believe the incident, in a packed Liverpool city centre on Monday, was isolated and not an act of terrorism.
They said the driver of the grey Ford Galaxy people-carrier involved in the incident was believed to have followed an ambulance into a closed street when a road block was lifted to allow paramedics to attend to a suspected heart attack victim.
Videos posted online showed the vehicle driving through the street crowded with fans, sending several flying into the air and dragging at least four under its wheels.
When the vehicle stopped, angry people converged on it and began smashing the windows as police officers battled to prevent them from reaching the driver.
Police said 50 people, including children, had been treated for their injuries, with 11 still in hospital.
"They are all in a stable condition, and I am pleased to say that they appear to be recovering well," Deputy Chief Constable Jenny Sims said.
King Charles, who is visiting Canada, said he was "deeply shocked and saddened to hear of the terrible events".
His sister, Princess Anne, met medics who had treated some of the injured at the Royal Liverpool University Hospital.
With most people off work for the Spring Bank Holiday, officials estimated that around one million people descended on the 16km parade route to watch the Liverpool team travel through the city on an open-top bus with the Premier League trophy.
Liverpool last won the league in 2020, ending a 30-year-wait, but fans were unable to celebrate due to lockdowns during the pandemic.
Police said the car hit the spectators as the event was winding down. In the aftermath, a Reuters photographer saw emergency services carrying victims on stretchers and in their arms to nearby ambulances.
One source told MailOnline that it looked like the driver panicked when he realised he was in the crowd and people started banging on his car.
The driver, who was sounding his horn, reversed and then accelerated forwards, according to reports from other witnesses.
Police were unusually quick to provide a description of the man they arrested, saying around two hours after the incident he was a "53-year-old white British man from the Liverpool area."
Former police officers and local politicians said that statement was needed to cool social media speculation that the episode was an Islamist attack.
"That was one of my first concerns, that we needed to get the story out quickly," Mayor of Liverpool City Region Steve Rotheram told the BBC.
"If there's a vacuum, we know there are some elements that will try to inflame the situation and to create that speculation and to put misinformation out there."
The same police force oversaw the response to the murder of three young girls in the nearby town of Southport last year, an incident which sparked days of rioting, fuelled initially by speculation online over the identity of the attacker.
