"Yes, people would argue that the 2014 tournament was a major disappointment," Neville said in an interview in Henry Winter's book "Fifty Years of Hurt".
"I'd agree with that but this summer is a big moment in terms of how the four years will be viewed, in terms of the four-year contract we were all given. People will judge it based upon tournaments."
The Football Association decided not to discuss a new deal for Hodgson until after Euro 2016, which begins in France on June 10, but Neville said the 68-year-old deserved an extension based on what he had achieved so far.
"I think Roy can be proud of an excellent piece of work. It needs a good finish in the summer," Neville added.
"My view would be that he would go on beyond the summer because at this moment in time he's doing the right things and making the right decisions, not just for himself, but for the team long term.
"He's made a series of really good decisions for the English team but also for future tournaments, for the next two-to-four years, and hopefully he, we, can see that through."
England, who play Portugal in their final warm-up match at Wembley on Thursday, face Russia, Slovakia and Wales in Group B of the 24-team tournament in France.
(Reporting by Ian Rodricks in Bengaluru; Editing by John O'Brien)
