The England defender’s father was caught in the chaotic scenes inside the stadium on Sunday night and struggled to breathe.
Maguire described what happened as “scary” and said her family will be “more aware” when they attend games in the future.
The 28-year-old, who flatly smashed his penalty kick in England’s shootout loss, said The sun: “My father was in the rush. I didn’t talk to him too much but I’m glad my kids didn’t go to the game.
“It was scary – he said he was scared and I don’t want anyone to go through that at a football game.
“I saw a lot of videos and talked to dad and my family. It was my father and my agent who suffered the most. He had trouble breathing afterwards because of his ribs, but he’s not the type to make a big deal out of it – he did.
“I hope we can learn some lessons and make sure it doesn’t happen again. Dad will always support me and go to the games, but he will be a little more aware of everything that is going on around. We should all be more aware and learn from it. “
“Things could have been a lot worse, but we have to make sure that doesn’t happen again. “
On Tuesday, UEFA opened a disciplinary investigation into the incidents of the Euro 2020 final at Wembley.
Football Association chief executive Mark Bullingham said on Monday that a full review would take place into what had happened.
He accused “drunken drunks” of trying to break into the stadium without tickets.
The FA has been indicted by UEFA on four counts stemming from the chaos. Disciplinary proceedings were initiated for an invasion of land, throwing objects, boos from the Italian national anthem and the lighting of fireworks.
Maguire is hopeful that what happened on Sunday night won’t affect England’s chances of hosting a future World Cup.
He said: “Wembley is an amazing place. It’s been an amazing house and the fans and the way they acted have been amazing other than a few rare in a final.
“We have to learn and we will – and it would be an amazing place to host the World Cup. “