Lessons must be learned from a security lapse which allowed protesters to smash the Prince of Wales' car window, Prime Minister David Cameron has said.
Prince Charles, who was in a Rolls-Royce on his way to the theatre with Camilla, said he "totally understood" the difficulties police faced.
The Met Police is to launch a major criminal investigation into student disorder in London over the last month.
MPs pushed through plans to raise the maximum tuition fee level to £9,000.
But 21 Lib Dems voted against the proposals, slashing the government's majority.
Thousands of students had gathered in London ahead of the vote before the demonstration turned violent.
Protesters largely took over Parliament Square and pressed against police lines in front of the Houses of Parliament.
Mounted police were used to control crowds, at one point charging a group of protesters. Hundreds were contained on Westminster Bridge for a time by officers.
Protesters threw flares, sticks, snooker balls and paint balls, according to police. After nightfall, riot police forced back protesters who were smashing windows at the Treasury and the Supreme Court.
Then at 1915 GMT, protesters launched an attack on the royal couple's car as it passed along a busy Regent Street. The Rolls Royce was kicked, splashed with paint and a window was smashed to chants of "off with their heads".
Met Police chief Sir Paul Stephenson called the attack shocking and said the couple should be commended for their fortitude.
He also said the route was thoroughly checked up to the last minute and his officers had shown "commendable restraint".
He said royal protection officers were put in a difficult position by the unpredictability of the "thugs" and said there would be an internal investigation.
Protest organisers from the Education Activist Network said there had been no call to attack the prince's car.
Mr Cameron called it a "very regrettable lapse of security" and said lessons must be learned.
"Let's remember that this was not the fault of the police," he added.
"Responsibility for smashing property, for violence, lies with the people that perpetrate that violence and I want to see them arrested and punished."
Clarence House has refused to comment on reports that the Duchess was poked with a stick, but did say the royal couple were unharmed and attended the Royal Variety Performance at the London Palladium as scheduled.
A former head of royal protection, Dai Davies, said he was surprised by the apparent lack of strategy.
"One of the principles of protection is to have alternative routes and I would have expected there to be at least three different routes," he told BBC Radio 4's Today programme.
London Mayor Boris Johnson said it was regrettable the heir to the throne could be surrounded by agitators and that people's first instinct was to blame the police.
Labour leader Ed Miliband joined the condemnation, saying the attack was "totally unacceptable" and the wider vandalism was inexcusable.
Students have criticised police tactics, particularly of holding demonstrators in a small area, known as "kettling".
Simon Hardy, of the National Campaign against Fees and Cuts, said: "The violence we witnessed yesterday from the police, before anything had been done by the protesters, is typical of how the Met has responded to these demonstrations.
"They turned up in riot gear, with horses, batons and shields. They said they wanted people to leave to Victoria Embankment and then 'kettled' them and then started beating people.
"Anyone who tries to resist is then told they are being violent."
But the Met Police Federation said officers acted with professionalism and selflessness and that, if they had not, the consequences would have been "unthinkable".
The Independent Police Complaints Commission (IPCC) is investigating a claim that 20-year-old Middlesex University student Alfie Meadows suffered serious head injuries after being hit on the head with a police truncheon. The Met Police referred the incident to the commission, which is appealing for witnesses, early on Friday.
The IPCC said it received three further complaints about the Met Police at Thursday's protests.
Further developments on Friday included:
- Scotland Yard updated figures of the injured to 12 officers and 43 protesters and said of the 33 people arrested, most were released on bail and no-one has been charged.
- The Met launched an investigation into student disorder seen across London between 10 November and 10 December.
- A small but vocal demonstration of about 100 people chanted about tuition fees as the prime minister arrived in the Harehills area of Leeds for a speech.
- Don Foster, one Lib Dem MP to vote for the rise, has had a rock thrown through his office window.
- Cambridge University student Charlie Gilmour, son of Pink Floyd guitarist David Gilmour, apologised for climbing the Cenotaph, a memorial to Britain's war dead, calling it a "moment of idiocy".
- Business Secretary Vince Cable said the Lib Dems were still "united" despite the rebellion.
Only 28 Lib Dem MPs - fewer than half - voted for the government's plans for tuition fees. Six Conservative MPs voted against. Three ministerial aides resigned.
The package of measures will see fees rising to an upper limit of £9,000 per year - with universities charging more than £6,000 per year told to protect access for poorer students.