The final finished 1-1 and went to penalties, where Harry Kane, Harry Maguire, Marcus Rashford, Jadon Sancho and Bukayo Saka stepped up to take the spot kicks, with the latter three missing theirs to hand the title to Italy.
Grealish, who was brought on as a substitute, hit back at comments from former Manchester United and Ireland player Roy Keane that he should have taken a penalty.
"I said I wanted to take one," Grealish posted on Twitter on Monday. "The gaffer [Gareth Southgate] has made so many right decisions through this tournament and he did tonight!
"But I won't have people say that I didn't want to take a [penalty] when I said I will."
Keane said teenager Saka should not have been given the task of taking England's crucial fifth penalty.
Saka, 19, had to score to send the shootout to sudden death, but his kick was saved by Gianluigi Donnarumma. The Italian goalkeeper had also saved from Sancho after Rashford had hit the post.
Keane said more experienced players should have stepped up and taken the responsibility.
"If you're [Raheem] Sterling or Grealish, you cannot sit there and have a young kid [Saka] go up for a penalty ahead of you," he told ITV.
"They have a lot more experience, Sterling has won trophies, they had to get in front of the young kid and stand up."
Roma manager Jose Mourinho echoed Keane's comments.
"For Saka to have the destiny of a country on his shoulders, I think it is too much for a kid to have everything on his shoulders in this moment," Mourinho told talkSPORT.
"But I do not know if I have to ask that question to Gareth or not. Because many times what happens is that players who should be there, they are not there. Players who should be there, they run away from responsibility."
Former England defender Gary Neville said the decision on who was taking the penalties would have been made well before Sunday's final.
"They would have worked out over the last few weeks in camp, done sessions on it, looked at who's scoring the most and got the best record," he said. "It would be scientific, it would be data-led."
Former England defender Rio Ferdinand said no one could "blame and point the finger" toward the players who missed the penalties, while Alan Shearer said players should be praised for stepping up.
"For the youngsters to say, 'Yes, I'll go on and take a penalty' ... you have to give them credit for that," former England skipper Shearer told the BBC. "But it will be tough for them now. Hopefully, they will get over it."