England suffered the ultimate heartache as they were beaten in a spot kick contest after a tense final ended in a 1-1 draw at Wembley,
with Keane suggesting Arsenal youngster Bukayo Saka should not have been asked to take a penalty.
"If you're (Raheem) Sterling or Grealish, you cannot sit there and let a young kid walk up ahead of you," he told ITV Sport.
"You can't sit there, see a young kid at 19 who is shy walking up in front of me, and I've played a lot more games and have a lot more experience?
"Sterling has won trophies. I'm not saying he wasn't prepared, maybe Gareth had him sixth or seventh.
"You can't sit there, that must be hard to take. You pull this young kid and say 'listen, I'll step up for you."
Now Grealish has rejected Keane's claims, as he suggested he wanted to take a penalty in the shoot-out, but was denied the opportunity by Southgate.
Jose Mourinho has also given his view on the Saka penalty decision, as he suggested the England management may have made a mistake to asked him to take the fifth penalty.
"The decision of the penalty takers, I think it is hard to leave Saka as the last one," Mourinho told talkSPORT. "I think it is hard for a kid to have everything on his shoulders at that moment. I just feel very sorry for him.
"In this situation where was Sterling, where was (John) Stones, where was (Luke) Shaw?
"I feel that Gareth is such an honest guy. Such a protective coach of his players. I don't think he would ever say players were not ready (to take a penalty)."
Despite the heartbreak of losing the Euro 2020 final on penalties, Mourinho believes England can take confidence into next year's World Cup in Qatar.
"I believe that if you look forward and you know that you have a World Cup coming very soon there are only reasons to be optimistic. Move to the future with great hope. A lot of these players can be even better for the experience.
"I think people have to start looking at the England national side with different eyes and, for the next World Cup, England will be a strong contender.
"But at the same time I can imagine the frustration and sadness because it was closer than ever. To lose a final at home is very, very hard.
"I think it is a missed opportunity. When you get to a final, anything less than a win is not good. When you lose a final I don't think you ever forget that. It stays with you forever. I don't know when they are going to sleep properly because it is hard.
"But the reality is that they did very well. They have a very young team. Gareth and Steve (Holland) together are doing a great job."