Cricket is no longer a sport where things are and can be hidden. Its a sport that is watched by millions and is also followed by millions. In fact there is battle going on as to whether cricket needs to be included in the "RTI act of India" so that the public gets to know what is happening. Obviously when you remain quiet, you are perceived to be hiding something and that is looked at negatively. Kumble being a man of principles and clean nature, decide to put out his thoughts in the open and yet he was very diplomatic in the way he worded his exit letter. He didn't point fingers at anyone and yet he said that due to the "reservations of the captain", it is not possible to continue and it is best to move on.
Kohli can do a similar thing where he puts his side of the story diplomatically. Anil Kumble is a hugely respected figure in international cricket, not just Indian cricket. Apart from a few handful of Kohli supporters, there is a huge number of cricket followers who have sided with Anil Kumble (its clear to see, if you scout social media and forums, etc.). It would be good if Kohli clears his side up and gets on with his life. Well if he does not decide to, then he should face the brickbats. If he is right in this whole saga, let him put his argument forward.