Roy Hodgson's side bowed out of the tournament after losing a penalty shoot-out to Italy last Sunday, meaning it is now 16 years since England progressed beyond that stage at a major competition.
Shearer was part of the team that reached the semi-finals at Euro 96 under Terry Venables and he admitted he was disappointed with the way the current squad performed in Poland and Ukraine.
"If we accept this level of performance, then we have given up on England," Shearer wrote in his column for The Sun.
"Have we dropped so far down the pecking order that we are applauding England for trying hard and being well-organised? That is the least you should expect from any side.
"I cannot remember the last time the nation celebrated the team getting out of our group. That used to be the least that was expected. Now it seems to be a pointer that we are heading in the right direction.
"Well, we are not. We are getting progressively worse if you look at our performances in the last three major tournaments we qualified for."
England finished top of Group D after victories against Sweden and Ukraine but were comprehensively outplayed against the Azzurri. And Shearer suggested that the number of foreign players in the Premier League has had a detrimental effect on the national team.
"I wonder why England were so poor," he said. "There’s a goalkeeper and three defenders who shared title and Champions League medals between them from last season, another title winner in James Milner, an exceptional midfielder in Gerrard and a world-class striker in Wayne Rooney.
"Yet put us all together and what do we get? Players who cannot pass the ball to each other. Perhaps the problem is we have too many foreign players in the Premier League and not enough English talent gets through.
"Perhaps that is the price we pay for having a league as diverse and entertaining as ours. Perhaps the FA have one plan, the Premier League another and until there is one cohesive idea to get a better national team things will not change.
"Perhaps England is not important enough to everyone any more. Which is why we go home with our heads held high along with the Czechs and Greeks. Maybe that’s just where we are at."