The 19-year-old, who will start life in the youth squad at the Britannia Stadium, joined the Potters for a pre-season trial before penning terms.
"The competition at Feyenoord is enormous," Gomez told De Telegraaf. "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."
Gomez is Stoke's fourth signing of the summer, while Juan Agudelo is expected to join from New England Revolution in January.