Calvin does not treat the issue directly, but seems to understand that by healing, restoration from a sin is meant. This is the interpretation that I am immediately inclined to. However, a strong counter-argument can be made from the context.
13Are any of you suffering hardships? You should pray. Are any of you happy? You should sing praises. 14Are any of you sick? You should call for the elders of the church to come and pray over you, anointing you with oil in the name of the Lord. 15Such a prayer offered in faith will heal the sick, and the Lord will make you well. And if you have committed any sins, you will be forgiven. —James 5:13-15
There seems to be a connection between sin and sickness already in verse 15. Henry comments,
Where sickness is sent as a punishment for some particular sin, that sin shall be pardoned, and in token thereof the sickness shall be removed.
I am not sure how this square with the teaching about sin ad suffering elsewhere in Scripture, but the most natural reading of this passage itself it to keep that connection in mind when interpreting 16: when suffering from a sin-induced sickness, confess and pray with brothers, and healing will be given.
Gill says with uncharacteristic brevity,
And pray for one another, that ye may be healed; both corporeally and spiritually.
If this interpretation of the passage is taken, then its immediate application is much narrower than the typical (out-of-context) way I've heard it used.