It makes more sense in the book, because Claire left Frank's ring on during the ceremony and that's why she wore Jamie's father's ring (and later her silver wedding band) on her right hand.

In the show there may be a slight hesitation on Claire's part and that's why Jamie chooses her right hand, not wanting to push Claire too much, knowing she had been wearing her gold band on the left. But it's not an obvious choice, so I think it's just one of those things that got poorly translated from book to screen.