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 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."
The League Manager's Association added: "The LMA is able to confirm that Tony Pulis has left Stoke City on amicable terms."