Walters, 29, failed to score from the spot as the Potters crashed to their fifth defeat in seven matches. Australian keeper Mark Schwarzer managed to save the Republic of Ireland international's spot kick, with the striker having scored just five goals this season.
"People have asked me that, and people say 'we would get him to take it, or we would get him to take it'. That's all fine, but do those players really want to take it when the game is going on?” Pulis told reporters.
"Jon picked that ball up [in the Fulham match] and there was no doubt about it - he wanted to take the penalty.
"He missed it, and as I have said before, people do things in life sometimes that are not right, or do not go the way they or you want them to go. But that is life, and you have to get on with it."
Stoke entertain West Ham at the Britannia Stadium on Saturday afternoon, and Pulis believes that Walters is still the right man for the job if they win another penalty.
"You have to make sure you back the players who are prepared to do it," he added. "If we get a penalty on Saturday and Jon picks the ball up and wants to take it - and there is nine or 10 others who don't - then let Jon take it."