Hell Indonesia alone has over 17,000 islands. I don't think it's even known who many could support human habitation ,let's not forget trade, and there is a world of difference between having a look at an island through a sat image passing over and actual exploration.
Nature is diverse. It's amazing to see what people eat and when you get down to it, the sea can provide an abundence of food. That people used to live this way for thousands of years, should be a good indicator that it's only a matter of understanding how to.
That it's not possible to sustain yourself off of every little speck of dirt poking up through the surface of the oceans, I would suggest that should you look, hard enough, you could find one.
Indonesia has 17,000 islands and 11,000 of them are inhabited. I suggest that the rest of them are unfit for habitation. Remember your original premise was to find an island and "drop off the face of the Earth and never see another person for the rest of your days" In my estimation that would be pretty hard to do if not impossible.