The Three Lions exited the tournament at the group stages for the first time since 1958, finishing bottom of Group D without a win and with just a solitary point to their name.
Roy Hodgson's side began their Euro 2016 qualifying campaign on a high with a 2-0 victory over Switzerland in Basel on Monday, but Owen believes that it will take a good performance in a major tournament to lift the gloom.
"They're only human. When results aren't going their way they probably feel it more than anyone," Owen said at the Soccerex conference in Manchester.
"They're in the 100-degree heat having to chase around. It's a painful experience, it can leave a scar on you.
"I think England need a good qualifying campaign and a big tournament."
Owen scored 40 goals in 89 appearances for England between 1998 and 2008.