The striker is joining the Toffees for a fee of €32.5m, having been on Merseyside since Tuesday.
Barry, who joined Villarreal a year ago from FC Basel for €14m, posted: "It has been an immense honour to have worn this shirt.
"Thank you for the love you've given me since day one, with that beautiful memory of my first goal in the stadium. That's what gave me the strength to go out and fight for you every weekend.
"I want to thank my teammates for giving me an incredible year, full of joy and the wonderful people I've met within the club and the team."
He also stated: "I thank the manager for having trusted me from the first call we had , which convinced me to come to Villarreal. Thanks to him, I have progressed as a player and as a person, and now I even speak a little Spanish!
"It's been a crazy adventure that has led us to qualify for the Champions League, and I hope the team goes far in this competition."