I think there are passages in the Scripture that speak of the healing of spiritual sickness, though admittedly they are in the minority. This seems to be one of those things where--as with theosis--the early Christians saw gems in the biblical text and polished them up for us. Regarding your questions, the most important passage IMO would be…
“And when the scribes and Pharisees saw him eat with publicans and sinners, they said unto his disciples, How is it that he eateth and drinketh with publicans and sinners? When Jesus heard it, he saith unto them, They that are whole have no need of the physician, but they that are sick: I came not to call the righteous, but sinners to repentance.” - Mk. 2:16-17
Obviously Jesus Christ is not talking about physical illness when he mentions a “physician” and people being “sick,” but rather, when he speaks of “they that are sick” he is speaking of “sinners”. Matt. 9:11-13 and Lk. 5:30-32 echo this passage in Mark.
The other way of looking at that passage - Mk. 2:16-17, is that Jesus was criticizing the Pharisees, since there are none who are righteous ,and all need our savior Jesus Christ. So he was being facetious when he specifically told the Pharisees that he did not come to call the "Righteous". Because they thought they did everything right and did not think they needed saving by anyone, especially Jesus who they thought was the one who needed them.
There is another thing that Christ said that would address the question being asked here , and that is in John 9:2, when the disciples asked whether it was the blind man's sin, or his parent's sin that caused his blindness. Jesus then told them that it was all our fault essentially, as the man was made blind by God so that we all could see God's Grace when he was healed by Jesus.
So in that he was telling us that it is not sin that causes sickness, and the best interpretations I have seen by Orthodox writers is that God does not punish us for our sin. That is rese4rved for the judgement day, and he is patiently waiting for the prodigal son's return.