The Everton centre-back has joined the Championship club on loan for the 2018/19 season.
Rowett said: "We have two very good senior central defenders in Ryan Shawcross and Bruno Martins Indi but I felt it was important we brought in another experienced player to add to the competition for places.
"Having spoken to Ashley he feels like he has a point to prove and has a real hunger. He knows many of our players, knows what we're all about as a club and will fit straight in to our culture."
Williams becomes Rowett's sixth signing of the summer as the Potters look for an immediate return to the Premier League.