The German, 31, was adjudged to have broken radio transmission rules and was demoted from second to third place.
Mercedes were found to have gone beyond the degree permitted to help a driver solve a technical problem.
The team said in a statement they "accept the stewards' interpretations of the regulations and their decision".
Rosberg's one-place demotion meant his lead at the top of the drivers' standings was cut to a single point by team-mate Hamilton.
The Formula 1 rules were changed this year to introduce restrictions on how much teams can help drivers while they are out on track.
Teams are allowed to intervene if the car has a "critical problem if a failure of a component or system is imminent and potentially terminal".
Race stewards found Mercedes were allowed to give Rosberg instructions on the settings he needed to resolve the problem.
But they should not have responded to his subsequent question on whether he should then change gear so the car went straight through seventh gear.
Rosberg's engineer Tony Ross replied to the German's information that he had a gearbox problem with the words: "Driver default 1-0-1, chassis default 0-1, chassis default 0-1. Avoid seventh gear, Nico, avoid seventh gear."
Rosberg then asked: "What does that mean? I have to shift through it?"
Engineer: "Affirm Nico, you need to shift through it. Affirm, you need to shift through it."
Mercedes have withdrawn their appeal after driver Nico Rosberg was penalised by 10 seconds during the British Grand Prix, which was won by Lewis Hamilton.