That's certainly what two European sociologists, Loek Halman and Thorleif Pettersson, have concluded. Using data from the European Values Survey, they found that there was no relationship between how religious a country was (on average) and a rich it was in social capital. For example, the Czech Republic and Slovakia have similar levels of social capital, although Slovakia is far more religious than the Czech Republic. Some of the countries with the most social capital, Sweden and Denmark, were also the least religious. In fact, in Western Europe, the trend is the reverse of what you might expect - the least religious nations have the most social capital! Now, the important fact to bear in mind is that, in Europe as in the USA, more religious people are more civically engaged. It's just that, at the aggregate level, other factors are overwhelmingly more important. For example, social trust, a key generator of social capital, is driven at a cross-national level by the same factors that build a strong democracy - such as open institutions and free speech. Although religious are generally perceived to be more trustworthy on an individual level, that really has no bearing at a national level. In other words, this is another example where extrapolating from the individual effects of religion to the social effects just does not work.
I think the last sentence is way too strong. The problem with these research programs is that people have strong preferences on what the outcomes should be based on their normative framework. That is, whether you accept the truth claims of religion or not should have no necessary bearing on whether you accept or reject hypotheses in regards to religion's social utility, but operationally there are clear trends. Because of the nature of social science it isn't too hard to keep looking for the research which just happens to support your particular preferences. I suspect that arguments to the effect that only within nation comparisons are useful, or, that between nation comparisons are useful, will track the preferences much more closely than any a priori methodological bias. There is much more will to support particular conclusions than to be cautious about teasing apart social parameters. In any case, in the United States it is rather clear that distance from Canada is the primary variable when it comes to social capital formation.