Cook's leadership qualities have been questioned once again following the Test series defeat to Sri Lanka, while the 29-year-old has not scored a century since last May.
However, Stewart believes that Cook must avoid engaging with his critics, and instead concentrate on returning to the form which has seen him score 8,125 runs.
"If it upsets you, don't read it, or listen to it. If it doesn't bother you then carry on as you wish, but it spurred me on to prove people wrong," the 51-year-old, who played 133 Test matches for England, told The Mirror.
"Once you become high profile in whatever world you are in, you've got to take the rough with the smooth. There is no grey area. You either win or you don't. You either score the runs your team needs or you don't and that is the way it has always been.
"The best way to deal with it, I found, was to use it to motivate you to go out there and show people what you're all about - show them how good you are."
The five-match Test series begins against India on July 9.