Griezmann, 25, finished with the Golden Boot as the tournament's top scorer. However, the French striker failed to add to his six goals in the hosts' extra-time defeat at the Stade de France on Sunday.
The Atletico Madrid forward saw a first-half header wonderfully saved by Portugal goalkeeper Rui Patricio, before heading a much easier chance over the bar later in the game.
The final had been billed in some quarters as a battle between Griezmann and Real Madrid star Ronaldo, 31, with the winner expected to claim the Ballon d'Or award as the world's best player.
But Griezmann said that the prospect of winning the individual award was not something he had thought about prior to Sunday's game.
"No, it will come if it has to come," Griezmann said. "Am I still in with a chance of winning? Cristiano Ronaldo won the biggest competitions. I think that that's it -- it's done."
Ronaldo had already won the Champions League with Real this season, scoring the winning penalty in a shootout against Griezmann's Atletico in May. But the Real Madrid forward made little impact in the Euro 2016 final as he was taken off on a stretcher 25 minutes into the game after failing to recover from a rough early tackle by France's Dimitri Payet.
And ex-France international Christophe Dugarry said he thought at the time that Ronaldo's withdrawal with a knee injury was going to be the turning point of the match.
"Yes and I wasn't the only one," Dugarry told Le Parisien. "The French players thought so too.
"Unconsciously, they maybe believed that the hardest part was done. But no.
"Success was on Portugal's side. We understood that when we saw Eder's goal.
"He shot without looking at the goal, almost to get rid of the ball and without necessarily believing that he would score. And it went in. In the end, it's a super goal and a stroke of genius."