I can't comment for why he has not resigned. It's not the norm to resign though, in the instance he has no alternative lined up. There are I presume, financial implications for him if he resigned, and like it or not, be is perfectly entitled to act in his best interests, as he sees it. I know I would.
My sympathy for him is on a human level, that's all. I don't like to see someone so obviously struggling in a job that is absolutely beyond his ability. He has had me tearing my hair out for the past 18 months. I'm fully aware that he leaves Everton a very wealthy man and shouldn't take too long to get over it. Ironically one of the banners at the protest read "Nice man, terrible manager" and that encapsulates how I feel about him.
The point is though, that a group of people at Everton, if all works out as intended in the next few days/week, decided some time ago that Martinez was no longer fit for purpose to continue as Everton manager. Why that decision was not put into effect as soon as possible after that consenus was reached, is quite simply beyond me. I don't get how it would have unduly interfered with the appointment of a replacement, or why someone could not have come in as interim/caretaker manager for a few weeks and done a half decent job. It the season had more games to run, they would probably have had to act anyway, given we are falling like a stone and morale is lower than a snake's backside.