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.
His Wigan side scored 47 goals on the way to dropping into the Championship - a tally bettered by only Southampton and Fulham among the sides from ninth place downwards.
"We can't lose the good things that Everton had last season. If you look from a football point of view they were the strongest team at home," he added.
"I am very fortunate to have the players Everton has at the moment. It is an incredible starting point.
"The Everton team last season never really showed a weakness. What you want to bring is an extra dimension and extra competition for places.
"The players know how to always be in games, they always know how to compete and that is something we can't lose next season.
"But we need to bring in some new signings, develop that competition and give us different options on the field.
"Slowly we need to bring a couple of players who can help the players in the dressing room.
"It is a natural transition and the right time to do it is in pre-season, we need to work on many aspects but more important we don't lose the phenomenal things they did last season.
"In any club you want to go from window to window having a stronger side and that is what we will have to work hard to make sure we don't let this group of players down."
While Martinez was credited with some good tactical decisions at Wigan, switching to a 3-4-3 formation being his most obvious, he stressed he would not be making huge changes to Everton's style initially.
However, he did say he wanted his players to become more attuned to being able to offer more to the side.
"The players in the dressing room are internationals, they can work on certain aspects and we can always work on other areas," he said.
"There is a big difference between the style and the system you play.
"The modern footballer needs to be flexible and be able to play in different formations,
"I think the style will be similar to what Everton's has been over the last few seasons. Then on top of that we need to try to develop other ways of playing.
"We will work on that and make sure we have that flexibility."