The 55-year-old parted company with the Potters by mutual consent on Tuesday after seven successive years in charge, ending a decade-long association either side of one season spent at Plymouth.
Pulis had come in for increasing criticism from supporters for his 'long-ball', physical style and Coates decided after five years in the Premier League it was time for a change.
"I have enjoyed some wonderful times at Stoke City and nobody was prouder than me when, after 23 years of exile from top-flight football we gained Barclays Premier League status," said the Welshman in a statement.
"Over the past five years we have created history by being the only team in this club's 150 years existence to remain out of the bottom six in top-flight football for five consecutive seasons.
"Within our five-year period in the Barclays Premier League we have featured in four major cup quarter-finals, an FA Cup semi-final and final.
"Further, we have encountered, progressed and reached the later stages of a major European cup competition, only to lose out to Spanish giants Valencia.
"Peter, his family and I have enjoyed every inch of the journey; he has been truly a fantastic man to work with.
"Although I am disappointed I do understand what he means when he says the board feels a need to take the club in a different direction."
Pulis believes he leaves the club in a much healthier position than when he rejoined in 2006, mainly as a result of the progress made once they were promoted.
"Of course, having reached the top you then have to find a way to stay there and develop a strategy for a club that had no infrastructure in place to compete with the top Championship clubs we had left behind, never mind the top Premier League clubs," he added in a statement released by the League Managers' Association.
"During this great run of success on the pitch the club has also been involved in the building and development of a training ground that now compares to any in the Barclays Premier League.
"Also, the Britannia Stadium has been revamped and now hosts and looks after virtually full-houses of the club's great supporters every other week.
"In leaving I am very proud of what we have all achieved.
"My immediate staff led by David Kemp, my backroom staff and all the training ground staff have been absolutely wonderful.
"To the supporters who have, and rightly so, been congratulated for their outstanding support especially at the Britannia and finally to all the players who have worked under me during the seven-year tenure I give my thanks for what has been a wonderful adventure."