Rosberg has won all four races this season -- and the last seven in total -- while triple world champion Hamilton has been bedevilled by mechanical problems and currently lies 43 points behind the German in second place in the standings.
"Dear All, I want you to know how grateful I am for all of your support. I'd like to ask that you please trust in my team, as I do," Hamilton wrote on Facebook.
"This is my family. These guys have been the greatest, hardest working people for me, and that is why I am now 3x World Champion.
"Please don't put any more thought into my team doing anything unjust towards me, and understand that it would be in no ones best interest for that to be the case."
He added: "We've had the best 3 years together, and whilst it's not going to plan right now, all will unfold in its own time. I trust these guys 1000% and my mechanics are incredible, the best in the business. I respect them so please do the same.
"They are the guys that are going to make winning this championship possible. Thank you once again. #Love#TeamLHMERCEDES AMG PETRONAS."
At last weekend's Russian Grand Prix in Sochi, Hamilton started 10th on the grid following an engine failure in qualifying and then suffered another mechanical problem before finishing the race in second place.
Since winning at the Black Sea resort last year, Hamilton has won only once -- the following race in the United States where he clinched the title.
After Sunday's race, Mercedes team chief Toto Wolff said the conspiracy theorists suggesting his outfit favour Rosberg are "a bunch of lunatics".
"I want to ignore this bunch of lunatics who think we would want to harm a driver, who is our driver and who has been a double world champion for us," said Wolff.