Sir Paul said: "The images that have now been released raise obvious concerns and it is absolutely right and proper that there is a full investigation into this matter, which the Met will fully support."
Home Secretary Jacqui Smith said the inquiry needed to be completed as soon as possible.
There was no reason for the officer to push him down
The video, shot at 1929 BST at the Royal Exchange Passage on 1 April, initially shows Mr Tomlinson, who was going home from work and not protesting, walking away from a group of police officers.
Footage shows that he then received a two-handed push from an officer, landing heavily before remonstrating with the police.
Minutes later Mr Tomlinson collapsed and died of a heart attack, after walking to nearby Cornhill where he received first aid from police.
Mr King told BBC Radio 5 Live: "For the sake of the family here and his kids we just want justice ... you know, until everything does come out and we do get the evidence we need, we can't lay our father to rest."
Reacting to the video, he said: "You can clearly see that my dad, Ian, had his hands in his pockets with his head down walking away.
"So there was no reason for the officer to push him down. If he did do something wrong, then why not arrest him in the beginning?
"We want answers now. You know, it's only minutes after [that] Ian collapsed and had the heart attack."
David Howarth says the video is ''plainly an assault'' on Mr Tomlinson
A New York fund manager recorded the footage, believed to be the last showing Mr Tomlinson alive.
He said he came forward with the video because the vendor's family "were not getting any answers".
The Liberal Democrats are now demanding a criminal inquiry.
The party's justice spokesman, David Howarth, said the footage showed a "sickening and unprovoked attack" by police.
Daniel Sandford, BBC home affairs correspondent, said of the footage: "This is now going to raise some more serious questions about the police behaviour on that night.
"Why is it that one of the officers walks up to a man who appears to be walking away from him?"
Sometimes it isn't clear, as a police officer, who is a protester and who is not
Peter Smyth of the Met Police Federation
Peter Smyth, chairman of the Metropolitan Police Federation, said some physical confrontation was inevitable during a large protest.
He told Radio 4's Today programme: "On a day like that, where there are some protesters who are quite clearly hell-bent on causing as much trouble as they can, there is inevitably going to be some physical confrontation.
"Sometimes it isn't clear, as a police officer, who is a protester and who is not.
"I know it's a generalisation but anybody in that part of the town at that time, the assumption would be that they are part of the protest.
"I accept that's perhaps not a clever assumption but it's a natural one."
'Quickly and thoroughly'
The Guardian newspaper obtained the video and has handed it to the IPCC.
Jacqui Smith said the investigation must be done ''as quickly as possible''
Ms Smith said: "I'm glad the IPCC themselves called for further evidence in order to be able to do that investigation as quickly and thoroughly as possible.
"And if it identifies - and I can understand people's concerns - if it identifies the need for a criminal investigation, that then also needs to be pursued."
A spokeswoman for the IPCC said: "We have recovered video footage from a national newspaper last night. We are now in the process of analysing it, along with the other evidence we have obtained on the case."
The police have well-established powers to use reasonable force if they think there is a threat either to themselves or the public, but these are enhanced during a protest or riot.
The key concept is that of "reasonable force" - i.e. force that is in proportion to the threat faced either by the public, the police or property
Thus "reasonable force" may literally range from putting a hand on someone's elbow, to shooting them dead
The legislation governing police powers during demonstrations (mostly the Public Order Act 1986) must also be seen in context of human rights' legislation
Under European human rights' laws the police are required to actively protect the public's right to peaceful protest
Policing during a protest, therefore, is a negotiation between the rights of police to use reasonable force to protect the public, and their responsibility to allow peaceful protest to take place
LOCATION OF IAN TOMLINSON AND G20 PROTESTS IN CITY OF LONDON
1. Protests at Bank Junction
2. Protest outside RBS branch in Threadneedle Street
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