I doubt that. Trial by combat may be part of the civil law, but the Sparrows are operating by ecclesiastical law... they're like the GoT version of the Spanish Inquisition. (And Cersei didn't expect them.)
I think they would view trial by combat as an unjust relic of the aristocratic rule they want to do away with. The richer and more influential the accused, the better fighter they can come up with. And I doubt the High Sparrow would endorse the idea that guilt and innocence are determined by who has the biggest sword.
So far, nobody has been arrested for anything they're not actually guilty of. (Whether the laws themselves are just or not is a different matter.)