The incident happened in the nationalist Carrick Hill area of north Belfast following the loyalist Tour of the North parade on Friday evening.
Mr Kelly says he was trying to stop the police vehicle as it drove off with a young man who had been arrested. He was filmed being carried on the front of it for a short distance.
The incident, which also saw Sinn Féin minister Carál Ní Chuilín require hospital treatment on her arm, has been referred to the Police Ombudsman.
Speaking on Monday, DUP leader Mr Robinson said: "Anybody that views the video evidence will consider it to have been reckless and ill-advised.
"He felt he had the authority to instruct the police to stop."
No one should interfere with the police during the conduct of their duties.
Mr Robinson added: "Anyone who has any misgivings about how police handle the issue, there is a proper process to go through that does not include blocking police in the execution of their duty.
"That is a bad example to set and I hope it won't have a bearing on the rest of the parading season.
"Being an elected representative, and even being a member of the Policing Board, does not afford him any role in the operation of police or giving police instructions on the site."
Mr Kelly, who met the Parades Commission on Monday morning, said the incident had overshadowed alleged breaches of restrictions on the march.
"I don't like the way police handled the situation and let me make this clear, I have not made a criticism of all the police on the night," he explained.
"I think there was a saturation approach to Carrick Hill which I do object to, but outside of that, we are talking about an incident involving a small number of police, and it was bad policing."
Jonathan Craig of the DUP said that the PSNI have confirmed they will investigate the incident.
He said: "The police have confirmed that they are investigating a number of possible offences surrounding the events on Friday night and that a file of evidence will be sent to the PPS.
"It is disappointing that more than 48 hours passed before this was confirmed, but it is vital that the community know that no-one is above the law and that all incidents are investigated regardless of who may have been involved."
The Chairman of the Police Federation Terry Spence says the talks held in Cardiff earlier this year, to improve the relationships between the police and community leaders in Northern Ireland ahead of the marching season, have failed at the first hurdle.
Police officers are the meat in the sandwich. It is a thankless job trying to protect the public and trying to ensure that public order is maintained.
He said the PSNI must investigate those who were involved in punching and kicking the police vehicle as well as senior politicians who intervened after the arrest of a teenager.
"These matters are the subject of a Police Ombudsman investigation," Mr Spence added.
"I am in no doubt a parallel investigation will be launched. There has to be. Not only for those involved in public disorder but also the actions of some politicians."
UUP leader Mike Nesbitt accused Mr Kelly of "breaking the spirit" of Cardiff.
He said: "Like the rest of us, Gerry Kelly spent two days discussing the issues, at the end of which he signed up to a declaration that included support for the PSNI, the Policing Board and the Office of the Ombudsman.
"Following Cardiff, we all received the mobile telephone numbers for two Assistant Chief Constables, four Chief Superintendents and the Chief Executive of the Policing Board.
"That seems to me to offer an alternative to obstructing a police vehicle in a manner that suggests an offence under Section 66 of the Police (NI) Act 1998."
However, SDLP representative for north Belfast, Alban Maginness, told UTV he was "not impressed" by how police acted on Friday, claiming the arrest of the juvenile "reignited" tensions in Carrick Hill district following the largely peaceful parade.
"It was a very ugly situation with enormous potential for serious public disorder," he added.
"The arrest of a person was a mistake and the events involving Gerry Kelly and the police land rover was a dangerous situation."
Earlier at Stormont the DUP called for the Assembly to look into the issue. The party also said it will raise the incident at the Policing Board.