Roy Hodgson's men topped Group D with a narrow 1-0 victory over co-host Ukraine, thanks to a Wayne Rooney goal on his return from suspension, while France was condemned to second after losing 2-0 to Sweden.
The Manchester City goalkeeper revealed that the squad has paid little attention to what's been going on elsewhere but hailed the support of English fans out in Poland and Ukraine and back at home.
"We've come here as a very strong group of players and we haven't listened to a lot going on outside of us," Hart told the FA's official website. "We've listened to the support of those magnificent people tonight and the brilliant people back home who have all wished us well.
"We're just really proud."
England now has until Sunday to prepare for its quarterfinal with Italy, after the team avoided an arguably tougher tie with defending champion Spain thanks to France's defeat.
Hart has insisted that England is ready to take on whoever it comes up against, however, and will continue to expect to win matches despite the harder opposition the team will now face.
"I feel we can achieve what we came here to achieve," Hart said. "We came here to win games.
"We played France in the first game, which was a tough start and we felt we could've won that game. But we've won these last two games and we've done it in the group stages, so we'll take on whoever is in front of us next."
There was a significant moment of controversy during the clash with Oleg Blokhin's men when Marko Devic's shot beat Hart and was subsequently hooked away by John Terry, with replays showing that the ball had already crossed the goal line before the Chelsea defender made his clearance.
Hart was unsympathetic to Ukraine's complaints, however, pointing out that the referee's decision is final.
"I don't really care to be honest," Hart added. "The referee's there to make decision, and he's made his decision. It's 1-0. We won 1-0. That's it."
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