I didn't find a religion-based study, but here's an interesting article on the birth rate in the Philippines:
http://www.prb.org/Publications/Articles/2010/philippines.aspx

The article says that "Data from the most recent Demographic and Health Survey (DHS) show total fertility rate falling from 6.0 lifetime births per woman in 1970 to 3.3 children per woman in 2006." and "The use of family planning increased from 17 percent in 1973 to 51 percent in 2008." However "The DHS results also show that the mortality rate for children under age 5 decreased from 58 deaths per 1,000 live births in 1998 to 34 deaths per 1,000 in 2008."

The article says that the birth rate decreases in inverse proportion to the income level of women, so richer women have lower birth rates.

The government of the Philippines has had programs promoting population growth control since 1970, and it appears that the programs (and, I would think, the change in society in general) have had a significant effect.

The article says "Average fertility for Southeast Asia in 2010 is about 2.4. With its moderately high fertility, the Philippines population of 94 million is growing by 1.8 percent annually."

That being said, I agree with Steve that the Catholic Church should drop its opposition to birth control - but I still don't think its position has a significant effect on population growth. Religion is only one of a wide variety of factors that affect birth rate.

It seems clear to me that nations that have had a traditionally low life expectancy, have higher birth rates. I would think that when life expectancy gets longer, the birth rate might drop, but at a rate less than the rate of increase in life expectancy - which would result in an increase in population. That seems to be the case in the Philippines.

-Joe-

P.S. Sorry, Steve Shaw, but I find you to be myopic, rigid, and overly dramatic - not to mention insulting. It's no fun discussing anything with you, because you don't play fair.
And I quote:

Tiny babies are forcibly christened and, as they grow, are force-fed religion by the million in schools and herded to church services, so perhaps this utopia of free choice isn't as clear cut as you're making it out to be. The horrid Christingle is almost upon us!

Bullshit!