Correlation does not equal causation. There are plenty of variables other than diet that can more easily explain both an increase in prosperity and a decrease in childbearing. Examples: high paying jobs increase wealth but decrease the time/energy available to raise a child. Or the less children one has the more money you can save. I think diet plays a negligible role in the relationship stated in the question.

There are probably dozens of reasons including diet. I have noticed that some of the doctors first to jump on the paleo bandwagon have been fertility doctors. Diet makes a difference in fertility. HOwever, I think the main reason for decline in birth is the increase in specialization and city life. In an agrarian society, the kids pitch in starting at around 5 years old, feeding the chickens, watching the herds, helping with weeds, harvesting, etc. Kids are basically turned quickly into labor. Over the course of their childhoods, they pay their way and then some.

Whereas in industrialized society, they are expensive and most of them are too busy going to school, etc to give tons back to the household. THus the kids become an expense and having hoards of children is a drain on the household income. Families are more likely to have only a few children. And with current methods of birth control, we have that choice. I think that is the main reason birth has declined. Economics. -Eva

Some reasoning, if not outright answers:

First of all, there's no question that a calorie-rich but nutrient-poor diet will slow population growth in any organism; this is why brewers use yeast nutrient.

Second, women with more education tend to have less sex and/or more birth control. Those who are kept unschooled are typically socialized to believe that their social role is the production of heirs, farmhands, and the like. The more educated believe that a few well-educated offspring will be enough to cover their needs in old age, and often delay reproduction attempts until it's plumb too late.

Third, there are environmental problems beyond mere diet that can impact fertility. Certainly a dietary fix would help, but the ideal scenario would be to filter the heavy metals and toxic organic chemicals out of the oceans, land, and air. This is why some people's back-yard vegetables are less healthy than the ones grown on a proper organic farm.

Last, you can look at the whole problem as a system of multiple nonlinear constraints. Population will expand to the carrying capacity of the land. If all of Pottenger's cats die, then the local alley cats will keep on yodeling in the night eating their diet of rodents, large insects, and junk. Diet is but one constraint. At some points in the parameter space, diet will be the limiting constraint. At other points, it will be insignificant or meaningless.

Here in my corner of the west, I suspect that our food supply is not as nutritious as it should be. But at the same time, I expect that the very largest problem is that our misguided education system scares girls away from having children until they are simply too old. People have grown dependent on the schools for their socialization, and the schools teach birth control, but they fail to mention the significant drop in biological fertility around the age of 30.

Women today also have the right to work. That is as much curse as blessing. It doubles the size of the labor pool without doubling the demand for labor. This pushes the natural price of labor so low that most families require two incomes. The consequence is that women do not reproduce, lacking the energy and time. If we mean to promote fertility among the wealthy west, we need an FMLA that lasts four years for pregnant women, or else a return to the notion that women of childbearing age should not work unless forced by unfortunate circumstances.