It has been suggested in some quarters that the club should have set their sights higher than a Spaniard who got Wigan relegated after eight seasons in the Barclays Premier League.
The 39-year-old's landmark FA Cup final victory over Manchester City at Wembley last month, which secured the Latics' first major trophy, appears not to have earned him much credit among his detractors.
But Martinez is not worried by those who are underwhelmed by his appointment on a four-year contract, insisting it only provides him with more motivation.
"In football it is normal and you need to have a debate because everyone has an opinion," he said.
"I think it is healthy people are questioning my appointment.
"It is something which keeps you desperate to see what you can do and one day turn them around so they are happy with the work you've done.
"I need to answer all the critics with work and that is nothing different to what I've done in the past.
"The only thing I want is for every fan at Everton to be proud of what we are trying to do."
Martinez blamed debilitating injuries to his back four for Wigan's relegation, with his former side conceding 73 goals in 38 matches - the league's joint-worst record with 19th-placed Reading.
Conversely, Everton's defence is much stronger - they lost just one league match at home last season - and Martinez will be looking to use that as a foundation on which to build a more attacking game.