Glossing over dubious assertions like "China [is] far more advanced than India due to its mandatory one-child policy", what has been the staple of demographers for the past 50 years at least is the so-called demographic transition:
the well-established historical correlation linking dropping fertility to social and economic development
As for the reasons for the drop in fertility, contraceptives/abortion are indeed assumed to be among them. Supposedly, one of the few counter-examples bucking the trend of the demographic transition has been Romania when they banned abortion in the 1960s:
in other Eastern European countries at the time, Romania’s fertility was declining
during the 1950s and early 1960s, and in 1965 the total fertility rate
of 1.91 births per woman was slightly below the Eastern European average
of 2.25. Legal abortion was the dominant method of birth control, and in
1965 about four legal abortions were recorded for every live birth (Berelson
1979: 209). By decree in October 1966 the government introduced overnight
a pronatalist policy whose main feature was the abrupt imposition of severe restrictions on legal abortion (although other measures were also introduced
to encourage births and restrict the availability of modern contraception). In
1966 Romania’s total fertility rate rose sharply to 3.66, while in other Eastern
European countries it averaged 2.16. The scale of the fertility rise moderated
after a few years. Some authors maintained that the effect was no more than
transient, but Berelson (1979) argued that an effect persisted for at least a
decade, and subsequent data suggest that it may have lasted even longer. The
nearly perpendicular rise in the graph depicting fertility is shown in Figure 2
and is contrasted with trends in selected other Eastern European countries.
This case meets most of the ten criteria we propose for causation. It highlights
also a further possible support for causal inference, namely the reversal
of a previous trend. Note too that the policy lasted for over a decade, and so,
to some extent, did the elevation of fertility, although it is evident that alternative
methods of birth control—including resort to illegal abortion—were
increasingly used. Documentary evidence is available on the principal mechanism—
an abrupt decrease in the number of legal abortions (Berelson 1979).
The abolition of legal abortion in the Soviet Union in 1936 is an analogous
case, but the subsequent rise in the birth rate was less pronounced and lasted
for only a few years, probably because the measure itself was less strict, less
widely adhered to, and was followed soon after by war (Avdeev and Monnier
1995; Avdeev et al. 1995).
Here's the continuation of that graph for Romania from a more recent paper:
Alas I don't know any studies that have tried to evaluate the impact of that (1960's decision) on the Romanian economy.
The first source I quoted mentioned some general theories for the demographic transition, not necessarily easy to validate:
Causal explanations for the fertility transition are of two main types
(Casterline 2003). First are those that emphasize aspects of social and economic
development (e.g., industrialization, sustained economic growth). These explanations often focus on changes in the costs and benefits of
having children, and a frequent theme is the emergence of incompatibility
between high fertility and the achievement of people’s economic objectives.
The second type of explanation is essentially demographic. It sees sustained
mortality decline as the process that, by raising the rate of population growth,
puts households and the wider society under strain, so eventually causing a
broadly compensatory decline in fertility. These explanations need not be mutually
exclusive. Both types appear in the classic formulations of demographic
As for the status of the demographic transition causation-wise in general... it's far more complicated; to quote what I wrote on Skeptics in reply to a similar question, by quoting a fairly cited paper:
Even with the hindsight of 50 years it has not been resolved whether the demographic transition is a theory, a generalization, a framework for analysis or merely an 'idea'. Or is it an 'historical model, predictive model, or a mere descriptivte term'? The debate about thes status of transition theory continues to occupy a central place in demography.