The former Leeds, Newcastle, Real Madrid, Middlesbrough and Tottenham defender was snapped up by the Potters on a free transfer.
Woodgate, who has signed an initial 12-month deal, has endured a nightmare time with injuries in recent years and was released by Spurs at the end of his contract this summer.
But the 31-year-old did not find himself short of suitors, with his abilities when fully fit having seen him earn England honours in the past.
A return to Elland Road was mooted, but the chance to remain in the Premier League with the Potters held the greatest appeal.
"I decided to go for a fresh challenge and spoke to Tony Pulis a few times," Woodgate told The Sentinel. "He sold me the club.
"There were a few other clubs interested, but Stoke are in the Premier League, they reached the FA Cup final and got into Europe."
Woodgate has made just four first-team appearances in the past two seasons and underwent surgery which saw a metal insert placed in his groin to help support his injured pelvis.
The talented centre-back traces his problems back to Tottenham's 9-1 victory over Wigan in November 2009, but now has his sights set on forcing his way into Pulis' first-team plans.
He added: "I ruptured the ligament at the bottom of the pelvis which caused the pelvis to become unstable.
"I had numerous operations and finally got back last year, but because I hadn't got a pre-season under my belt I kept getting little niggles. It was a nightmare.
"But you have to be optimistic. The medical team at Stoke is really good and their record in the Premier League is second to none.
"I've got to take it one step at a time, and the most important thing is to try to get into their first team first, which won't be easy because they've got a strong squad."