It's a bit more varied than that. The modern Western wedding ceremony is largely a simplified, streamlined, and often secularized form of the weddings that would have been common in the Middle Ages. In ye oldene dayse, the ring was first placed on the index finger (invoking God the Father), then the middle finger (invoking God the Son), and then the ring finger (invoking the Holy Spirit). With the Trinity thus completed, the ring was left on the ring finger. There's a parallel here with Jewish wedding ceremonies (where the ring is initially placed on the index finger and moved after the ceremony... although in the Semitic tradition the ring also switches hands, since the ceremony involves placing it on the right hand). But since there is apparently a 14th century "script" from Cologne that avoids the index and middle finger (Trauung und Jawort by Schott covers this a bit but I haven't had any luck finding a digital copy).

Whether or not the ring remains on the left hand, though, isn't really a religion thing. While it's true that predominantly Orthodox countries move the ring to the right hand, the same can also be said of the German-speaking world (though, oddly enough, Switzerland doesn't seem to follow that tradition).

You can find a "script" of how the vows would have been exchanged in mid-16th century England here, starting on page 78. Unlike early weddings (or maybe just continental weddings), it skips right to the "put the ring on the ring finger" bit, as is common in modern Western weddings.

I can't even justifiably break this down by region. While wearing the ring on your left hand is a largely Western European thing (and thus extended to much of the colonial world), even that is problematic because tradition in Spain holds for the wedding ring to be worn on the right hand. But this tradition isn't even uniformly exported to Spain's old colonies: in Mexico, the wedding ring is worn on the left hand and the engagement ring on the right hand (so, the reverse of the German tradition, basically).

And once you get beyond what we'd typically define as the Christian world pretty much all bets are off. Wedding rings don't really have a basis in Islam and so there's no real convention there for Muslim couples. Tradition in China is that men and women wear the rings on different hands (left and right, respectively). And then there are individual countries where it's not even consistent and largely varies depending on which part of the country you're from (see: Belgium).

Wedding traditions are complicated. :|