The announcement was made following the Champions League draw, with Ronaldo pipping teammate Gareth Bale and Atletico Madrid striker Griezmann to the award.
But Ronaldo, whose Portugal defeated Griezmann's France in the Euro 2016 final, said the 25-year-old would've been a deserved winner and is certain he will claim the award at some point in his career.
After collecting his award, the Portuguese star said: "Griezmann deserved to win this trophy.
"He lost two finals so it was unlucky for him but I think he's a fantastic player, he's a good boy, he's my neighbour - he lives next to me - and he deserved it.
"But he will win for sure one day - I hope so."
When asked about his chances of claiming a fourth Ballon d'Or, Ronaldo said he owes much of his personal success to the endevours of his teammates.
He continued: "I hope to win the Ballon d'Or but let me enjoy first this trophy. It's amazing. I wish to win tonight and I'm so proud. I have to thank you to my team mates - Real Madrid and national - because they helped me a lot to win these trophies."
Real Madrid were, meanwhile, paired with Borussia Dortmund, Legia Warsaw and Ronaldo's home-town club Sporting when the draw was completed, with the player admitting he is looking forward to facing his old side.
"It's going to be special," he said. "It's the club that I started to play football and will be a very emotional moment when I'm going to play against them.
"But it's good, I am happy we are in the same group."
It is the second time Ronaldo has been named UEFA's Best Player in Europe while Lyon's Norwegian striker Ada Hegerberg won the women's award.
She beat competition from France's Amandine Henry and German midfielder Dzsenifer Marozsan.