Delap, who was one of seven players released this week, has made more than 200 league and cup appearances since arriving at the Britannia Stadium six years ago.
And the midfielder, who spent the second half of the 2012/13 season on loan with Barnsley, hopes the fans now get behind new manager Mark Hughes.
"The supporters were brilliant in my time at the Club," said Delap.
"When I broke my leg, I received loads of letters and I can't say enough about what they did for me during my time at Stoke.
"There's no one who deserves success more than the Stoke fans, apart from the Chairman and his family.
"Hopefully, they will all get behind Mark Hughes. I'd love to see them push on and hopefully break into the top half of the table.
"In my second game on loan, I broke my leg. The Club, rightfully so, could have sent me back to Sunderland and waited for me to get fit.
"But we'd shaken on a permanent deal in January and the Chairman, Chief Executive and gaffer came to see me in the hospital and said they would honour the agreement.
"I couldn't believe it - I'd only been at the Club a week and it made me desperate to repay and hopefully I went some way towards doing that."