And Football Association chief executive Martin Glenn said it was now “imperative” to find out why England were “brittle” at tournaments.
Hodgson, who resigned after Monday’s game, repeatedly questioned why he had to attend a news conference on Tuesday.
But he said: “One particularly bad game has caused a lot of damage to me personally and the team going forward.”
Hodgson added: “We have a major bridge to repair – had we played better last night that might not need repairing.”
England won just one of their four games at the tournament in France, qualifying second from their group to reach the last 16, where they suffered the humiliating defeat by a nation with a population of just 330,000.
Hodgson said there were no “magic answers” to explain England’s performance at the European Championship, where they also drew with Russia and Slovakia and beat Wales with a last-minute winner.
Glenn did not rule out appointing a foreign manager as Hodgson’s successor, adding: “We are looking for the best person, not necessarily the best Englishman.”
But he said the key thing to address was England’s regular failure, which has seen them reach only three semi-finals at a major tournament since they won the 1966 World Cup as hosts.
“We need to punch our weight in tournaments in a way that we have not done in 50 years,” added Glenn.