Kevin Pietersen and Shane Warne have both criticised Cook in the last week for his captaincy style in the series against Sri Lanka, and suggested his performances with the bat do not warrant a place at the top of the order.
Defeat on the penultimate ball of the second Test at Headingley on Tuesday gave the tourists their first series win in England and increased the pressure on the 29-year-old.
Cook should not take the comments to heart, according to Stewart, who captained England in 1998, and the Surrey coach advised him to work hard on his technique to start scoring runs.
"When you haven't won for eight Test matches it (the dressing room) is not going to be the happiest place in the world," he told Sky Sports.
"I don't think it will be disjointed in any shape or form, (but) I tell you what if Alastair Cook scores runs and England win all this chat will stop.
"He's in a very high profile job, when you take on that role you know you are in the spotlight, you are the front man for that team.
"The most important thing is he scores runs, no-one should just be in the team as captain, he is an opening batsman who has got a tremendous record behind him.
"He has been England's leading run scorer in recent times, he is a very special player.
"At the moment he hasn't been scoring the runs that he would want to score and that the team would want him to score, he's got to get back to run scoring.
"He knows he can do it, but he does need runs very soon."