Preventing Tooth Decay
I guess most people know what a tooth is all about.
Enamel on the outside, and when that they gets damaged somehow we know we can expect problems some time down the road. Well, in the picture to the right that I borrowed compliments of Stephan,
you get a good idea of what your tooth or molar looks like.
Teeth are made of four layers. Enamel is the hardest, most mineralized outer
shell. Dentin is another protective mineralized layer that's below the enamel.
Below the dentin is the pulp, which contains blood vessels and nerves.
The roots are made of cementum, another mineralized tissue.
When enamel is poorly formed and the diet isn't adequate, enamel dissolves
and decay sets in. Tooth decay is an opportunistic infection that takes
advantage of poorly built or maintained teeth. If the diet remains inadequate,
the tooth has to be filled or removed, or the person risks more serious
Fortunately, a decaying or broken tooth has the ability to heal itself. (!!!)
Pulp contains cells called odontoblasts, which form new dentin if the diet is
good. And of course the key word here is : diet.
Most people associate good dental care with brushing and flossing.
Teeth are able to heal themselves. That's how traditional cultures such as
the Inuit can wear their teeth down to the pulp due to chewing leather and
sand-covered dried fish, yet still have an exceptionally low rate of tooth decay.
It's also how the African Wakamba tribe can file their front teeth into sharp
points without causing decay. Both cultures lost their resistance to tooth
decay after adopting nutrient-poor Western foods such as white flour and sugar.
Here's what Dr. Edward Mellanby had to say about his wife's research on the subject.
This is taken from Nutrition and Disease:

"Since the days of John Hunter (the caveman) it has been known that when the enamel and dentine are injured by attrition or caries, teeth do not remain passive but respond to the injury by producing a reaction of the odontoblasts in the dental pulp in an area generally corresponding to the damaged tissue and resulting in a laying down of what is known as secondary dentine. In 1922 M. Mellanby proceeded to investigate this phenomenon under varying nutritional conditions and found that she could control the secondary dentine laid down in the teeth of animals as a reaction to attrition both in quality and quantity, independently of the original structure of the tooth. Thus, when a diet of high calci­fying qualities, ie., one rich in vitamin D, calcium and phosphorus was given to the dogs during the period of attrition, the new secondary dentine laid down was abundant and well formed whether the original structure of the teeth was good or bad. On the other hand, a diet rich in cereals and poor in vitamin D resulted in the production of secondary dentine either small in amount or poorly calcified, and this happened even if the primary dentine was well formed."

In other words, it's the mineral content of the diet, particularly calcium and phosphorus
The fat-soluble vitamin content of the diet, chiefly vitamin D and as Dr Weston Price already had discovered Vit K2 (pasture fed butter)
The availability of minerals for absorption, determined largely by the diet's phytic acid content (which prevents mineral absorption).

So Drs. Mellanby set out to see if they could use their dietary principles to cure tooth decay that was already established. They divided 62 children with cavities into three different diet groups for 6 months.
Group 1 ate their normal diet plus oatmeal (rich in phytic acid).
Group 2 ate their normal diet plus vitamin D.
Group 3 ate a grain-free diet and took vitamin D.
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." This study was published in the British Medical Journal (1932, 1: 507) and the British Dental journal. Here's Dr. Edward Mellanby again:

The hardening of carious areas that takes place in the teeth of children fed on diets of high calcifying value indicates the arrest of the active process and may result in “healing” of the infected area. As might be surmised, this phenomenon is accompanied by a laying down of a thick barrier of well-formed secondary denture... Summing up these results it will be clear that the clinical deductions made on the basis of the animal experiments have been justified, and that it is now known how to diminish the spread of caries and even to stop the active carious process in many affected teeth.

It was dr. Weston Price who almost a century ago discovered that something must be seriously wrong with our diet when he found societies where our western diet had pretty much passed them by , with no teeth decay whatsoever.
It was Weston Price who started doing some research on  this"strange phenomenon" where people on a completely "wrong" diet were healthier, living longer and had none of the ailments we were suffering from on our much healthier diet.
Dr. Weston Price also had success curing tooth decay using a similar diet. He fed underprivileged children one very nutritious meal a day and monitored their dental health. From Nutrition and Physical Degeneration (p. 290):

About four ounces of tomato juice or orange juice and a teaspoonful of a mixture of equal parts of a very high vitamin natural cod liver oil and an especially high vitamin butter was given at the beginning of the meal. They then received a bowl containing approximately a pint of a very rich vegetable and meat stew, made largely from bone marrow and fine cuts of tender meat: the meat was usually broiled separately to retain its juice and then chopped very fine and added to the bone marrow meat soup which always contained finely chopped vegetables and plenty of very yellow carrots; for the next course they had cooked fruit, with very little sweetening, and rolls made from freshly ground whole wheat, which were spread with the high-vitamin butter. The wheat for the rolls was ground fresh every day in a motor driven coffee mill. Each child was also given two glasses of fresh whole milk. The menu was varied from day to day by substituting for the meat stew, fish chowder or organs of animals.
Dr. Price provides before and after X-rays showing re-calcification of cavity-ridden teeth on this program. His intervention was not exactly the same as Drs. Mellanby, but it was similar in many ways. Both diets were high in minerals, rich in fat-soluble vitamins (including D), and low in phytic acid.
Price's diet was not grain-free, but used rolls made from freshly ground whole wheat. Freshly ground whole wheat has a high phytase (the enzyme that degrades phytic acid) activity, thus in conjunction with the long yeast rises common in Price's time, it would have broken down nearly all of its own phytic acid. This would have made it a source of minerals rather than a sink for them. He also used high-vitamin pastured butter in conjunction with cod liver oil. We now know that the vitamin K2 in pastured butter is important for bone and tooth development and maintenance. This was something that Dr. Mellanby did not understand at the time, but modern science has corroborated Price's finding that K2 is synergistic with vitamin D in promoting skeletal and dental health.

Yet, wheat comes with its own set own problems, glutens, lectins, etc.

If we were to design the ultimate dietary program to heal cavities that incorporates the successes of both doctors,could it maybe look something like this??:

Rich in animal foods such as meat, organs, fish, bone broths, full-fat pastured dairy (if tolerated) and eggs.

Fermented grains only; no unfermented grains such as oatmeal, breakfast cereal, crackers, etc. No breads except sourdough because they typically aren't made from fresh flour.
No nuts; beans in moderation, only if they're soaked overnight or longer in warm water (due to the phytic acid).

Starchy vegetables such as potatoes and sweet potatoes.

Moderate quantities of fruit, but no refined sweets.

Moderate quantities of well-cooked vegetables.

Sunlight, high-vitamin cod liver oil or vitamin D3 supplements.

Generous amounts of pastured butter.
No industrially processed food.

This diet would maximize mineral absorption while providing abundant fat-soluble vitamins. It probably isn't necessary to follow it strictly. For example, if you eat more mineral-rich foods such as dairy and bone broths, you can probably get away with more phytic acid. Or you might be able to heal cavities eating like this for only one or two meals a day, as Dr. Price demonstrated. (Stephan)

This diet would be specifically for healing cavities. It does not mean that you'd always have to avoid all the foodstuffs mentioned here. I still believe oatmeal dressed up with fruits, walnuts, and cinnamon is still a pretty good breakfast.
If you want to really add something extra to it, you could put a tablespoon of coconut oil in it.

Complete avoidance of simple carbs will go a long way to fight  cavities: no sugar, no white bread, no white pasta's.