Returning to his home shortly after 3.30pm, Clarkson refused to answer when asked if he was worried about being arrested.
Instead, he said: "All I would like to say is ... I wish people would leave Oisin alone because none of this was his fault."
North Yorkshire Police have spoken to potential witnesses to the fracas which cost Jeremy Clarkson his job at the BBC, saying the force has "a duty to investigate" what happened.
Clarkson's Top Gear contract has not been renewed by the BBC after an internal investigation found he attacked show producer Oisin Tymon after an angry - and lengthy tirade - in a hotel in the county.
In a statement, the force said they had been contacted by "concerned members of the public" asking what action they were taking.
A spokesman said: "No-one who was present at the hotel during the incident came forward to report an offence to the police. Nonetheless, we have a duty to investigate where we believe an offence might have been committed, and that is what we have been doing with this case.
"We have already spoken to some people who were in the hotel at the time, but there are still others we need to speak to, who could help us to determine exactly what happened.
"Also, as is usual in these circumstances, we have made contact with Mr Tymon through his lawyers, to ask him to speak to us so we can ascertain how he wishes to pursue this matter.
"When these routine inquiries are completed, we will be able to assess whether or not further action needs to be taken from a police perspective.
"Meanwhile, we'd like to reassure the public that the matter is in hand, and we dealing with it responsibly and proportionately."
North Yorkshire Police today said it had spoken to some potential witnesses over the Jeremy Clarkson incident in a county hotel, adding it had "a duty to investigate where we believe an offence might have been committed".
Police believe a man missing from home in Essex may have travelled to North Yorkshire.
Michael Bell, 57, was last seen wearing a red fleece, black anorak and blue jeans.
He is believed be driving a Black Nissan Note index EX11 EOO.
A photo gallery as soldiers from a regiment based at Catterick Garrison in North Yorks conduct the final Afghan medal presentation parade.Read the full story ›
Soldiers from a regiment based at Catterick Garrison in North Yorkshire are conducting the final Afghan medal presentation parade in Richmond Castle.
The medal parade marks the end of Operation Herrick in Helmand province in Afghanistan.
The 46 medals are being presented for service during Op Herrick in May – November 2014.
Cloud, rain and hill snow clearing through the morning, then becoming mainly dry with some sunny spells by mid-afternoon, along with an isolated shower. Windy and feeling cool.
Maximum Temperature 9C.
Join Paul Brand for the last edition of Around The House before the General Election.
Tonight, Paul meets more of the candidates who are hoping to make it to Westminster.
Around The House is at 10.40pm on ITV.
A look ahead at the forecast for the Tyne Tees region:Read the full story ›