The force said that it had “completed its routine inquiries” into the incident and would “not be pursuing this matter any further”.
It said: “Now that all the interviews are complete, we have properly established that there is no need for further police action.”
Oisin Tymon, the producer attacked by Clarkson, said last week he did not want to press charges against his former colleague.
Mr Tymon had his lip split in a 30-second-long assault on March 4 and had to take himself to hospital with his injuries. He was also shouted at by the former Top Gear presenter in a torrent of verbal abuse.
Clarkson reported the incident to the BBC five days later and was suspended by the broadcaster on March 10 before it was announced his contract would not be renewed.
The star received widespread public support in the aftermath of the fracas while 1m people signed a petition calling for the BBC to reinstate him.
But director-general Tony Hall said ”a line has been crossed” and that ”there cannot be one rule for one and one rule for another”.
It emerged last week that Clarkson will appear on stage with his former Top Gear colleagues, Richard Hammond and James May, later this year to fulfil a series of gigs planned before he left the BBC.
The gigs will be stripped of all BBC branding and content, and billed as Clarkson, Hammond and May Live.
That means the shows, which will take in venues in Australia, Norway, South Africa and the UK, will not be able to use clips from the show or feature The Stig.
The move is likely to spark rumours the trio intend to continue their careers together after Clarkson’s exit from the corporation, but sources close to the tour say it is just a matter of fulfilling commitments to the fans.
The future of the show’s other two hosts — Hammond and May — is unclear, with both men’s Top Gear contracts having run out.
Police are still investigating threats to kill Hall, reportedly linked to his decision to axe Clarkson.