The verb iaomai means to heal completely and thoroughly — to restore to full health and wholeness. It differs subtly from therapeuō (G2323, to treat/care for) in that iaomai emphasizes the completed result: the person is healed. In the NT it is used of Jesus' miraculous healings and, significantly, of the spiritual healing that flows from His atoning work.
Isaiah 53:5 LXX is the prophetic foundation: 'and by his wounds we are healed [iathēmen].' Peter quotes this directly in 1 Peter 2:24: 'by his wounds you have been healed [iathēte]' — applying the physical healing imagery to the spiritual restoration of sinners. Jesus uses iaomai in Luke 4:18 (quoting Isaiah 61:1) to announce His messianic mission: 'He has sent me to proclaim freedom for the prisoners and recovery of sight for the blind, to set the oppressed free' — healing as embodied proclamation of the kingdom. John 12:40 quotes Isaiah 6:10 as the reason for Jewish unbelief: their hearts cannot turn and be healed [iaomai] — indicating that spiritual perception and healing are inseparable. Hebrews 12:13 applies iaomai to communal life: make level paths 'so that what is lame may not be dislocated but rather be healed' — the church as a healing community.