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.
Another passage that seems to express the Orthodox view is found in the Psalms: “I said, Lord, be merciful unto me: heal my soul; for I have sinned against thee.” (Ps. 41:4) Now the problem with this passage is that there is physical sickness mentioned in the previous verse (v. 3). However, I would argue that this previous verse is not related to the healing mentioned in v. 4, and that they are separate thoughts.
Here are some other passages to consider…
“Ah sinful nation, a people laden with iniquity, a seed of evildoers, children that are corrupters: they have forsaken the Lord, they have provoked the Holy One of Israel unto anger, they are gone away backward. Why should ye be stricken any more? ye will revolt more and more: the whole head is sick, and the whole heart faint. “ - Is. 1:4-5
“A voice was heard upon the high places, weeping and supplications of the children of Israel: for they have perverted their way, and they have forgotten the Lord their God. Return, ye backsliding children, and I will heal your backslidings. Behold, we come unto thee; for thou art the Lord our God.” - Jer. 3:21-22
“For I am with thee, saith the Lord, to save thee: though I make a full end of all nations whither I have scattered thee, yet I will not make a full end of thee: but I will correct thee in measure, and will not leave thee altogether unpunished. For thus saith the Lord, Thy bruise is incurable, and thy wound is grievous.” - Jer. 30:11-12 (cf Mic. 1:9)