The 19-year-old right-back impressed during pre-season trials with Stoke, and has now agreed a two-year deal with the Premier League outfit.
He will join Stoke's youth squad, and is happy to have made the move from Feyenoord.
Gomez told Dutch paper De Telegraaf: "The competition at Feyenoord is enormous. I have to compete with defenders of the Dutch national team, so I know it would be very difficult in the near future to get a chance at Feyenoord. Stoke City was therefore a fantastic club at the right moment.
"I want to work hard to get some chances in the first team this season. After the first training I received compliments from coach and players.
"The club have placed me in a guest family and I am completely in the English way of life. It's a dream and a fantastic experience."