The allegation, made to police on May 23 this year, related to an incident that was said to have occurred in Tower Hamlets, east London.
Police said that there was “insufficient evidence to proceed” with the claim.
A press spokesman for Mr Evans confirmed the case had been dropped.
The spokesman for the former Top Gear presenter said: “The Metropolitan Police have confirmed that the complaint has been investigated, but it fails to meet the evidential test to warrant any further action. Case is dropped.”
The Metropolitan Police said officers from the sexual offences, exploitation, and child abuse command investigated and a man in his 50s was interviewed under caution on Thursday.
But a spokesman said: “The decision was made by the police that there was insufficient evidence to proceed.”
Earlier this month Mr Evans announced he was to leave his role as a presenter on BBC Two’s Top Gear after one series, saying he had given it “my best shot but sometimes that’s not enough”.