Received public relations wisdom is you get on top of the story. But in this case, I think it only makes a private matter public. The current speculation is that he had an affair. What is he going to say in a statement that is going to change that opinion? What if he had an affair, denies it and later we find out the truth?
There is nothing to be gained by coming out with a statement other than he ran into a fire hydrant, which amounts to minor traffic accident.
It's the silence everyone resents. The media abhors a vacuum and in the absence of real news, everyone writes about the absence of real news.