Oisin Tymon, 36, suffered swelling and bleeding to his lip after being physically and verbally attacked by Jeremy Clarkson, 54, in a hotel in North Yorkshire.
He was also subjected to a prolonged verbal tirade and drove himself to hospital believing he had lost his job.
The incident saw Jeremy Clarkson dropped by the BBC after an internal inquiry.
In a statement Oisin Tymon’s lawyer, Paul Daniels, said: “Mr. Tymon has informed the police that he doesn’t want to press charges.
“The events of the last few weeks have been extremely unpleasant for everyone involved. The matter has taken a great toll on Oisin, his family and his friends.
“Quite simply, Mr. Tymon just wishes to return now to the job at the BBC he loves, as soon as possible.
“Further, the BBC have, in his view, taken action with a view to addressing the issues at hand.
“Mr. Tymon agrees with the BBC’s stated view that all parties should now be allowed to move on, so far as possible.”
Oisin Tymon, who previously described Jeremy Clarkson as a “unique talent”, has come under sustained abuse on social media for his involvement in the dispute.
Speaking outside his home on March 26, Jeremy Clarkson asked fans to show restraint.
“I wish people would leave Ois alone because none of this was his fault,” the presenter told reporters.
Oisin Tymon never lodged a formal complaint with the BBC following the altercation with Jeremy Clarkson, which took place on March 4.
The internal investigation was prompted after the presenter reported himself to Danny Cohen, the corporation’s director of television.
The investigation found that Oisin Tymon was subject to an “unprovoked physical and verbal attack”.
The physical altercation lasted “around 30 seconds and was halted by the intervention of a witness,” said the report’s author, Ken MacQuarrie.
As a result, the BBC’s director general, Tony Hall, said he would not renew Jeremy Clarkson’s contract.
“For me a line has been crossed,” Tony Hall said on March 25.
“There cannot be one rule for one and one rule for another dictated by either rank, or public relations and commercial considerations.”
Tony Hall said he “not taken this decision lightly” and recognized it would “divide opinion”.
Following Oisin Tymon’s decision not to press charges, North Yorkshire Police said the force was still investigating the incident.
“We are grateful to Mr. Tymon for clarifying his position, as we can now take this fully into account,” said a spokesman.
“We have some final interviews to complete with guests who witnessed the incident as their perspectives also need to be considered.
“When these conversations are complete, we will issue a statement on our website as appropriate.”
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