Can diet heal cavities?

During the last year, I have been quite focussed on my diet, but really with an eye toward weight loss than larger nutritional issues. Of course, this means I am eating a LOT more vegetables and fruits, and have been feeling better. But it never occurred to me that diet could affect my dental health (with, of course, the exeption of more sugar meaning more cavities). But at least one doctor is claiming that a diet low in grains and high in calcium can reverse tooth decay.

In group 1, oatmeal prevented healing and encouraged new cavities, presumably due to its ability to prevent mineral absorption. In group 2, simply adding vitamin D to the diet caused most cavities to heal and fewer to form. The most striking effect was in group 3, the group eating a grain-free diet plus vitamin D, in which nearly all cavities healed and very few new cavities developed. Grains are the main source of phytic acid in the modern diet, although we can't rule out the possibility that grains were promoting tooth decay through another mechanism as well.

Dr. Mellanby was quick to point out that diet 3 was not low in carbohydrate or even sugar: "Although [diet 3] contained no bread, porridge or other cereals, it included a moderate amount of carbohydrates, for plenty of milk, jam, sugar, potatoes and vegetables were eaten by this group of children."

I am skeptical, but it seems like something worth looking into.