"When the even was come, they brought unto him many that were possessed with devils: and he cast out the spirits with his word, and healed all that were sick: That it might be fulfilled which was spoken by Esaias the prophet, saying, Himself took our infirmities, and bare our sicknesses." (Matthew 8:16-17)
The earthly healing ministries of Jesus are here said to have fulfilled the prophecy of Isaiah 53:4 : "Surely he hath borne our griefs, and carried our sorrows." The Hebrew words used do allow this New Testament application, so this passage does, indeed, predict the healing work of the Messiah in His earthly ministry. Isaiah 53:5-6 then predicts the substitutionary atoning work of the Messiah, concluding with the statement: "The Lord hath laid on him the iniquity of us all."
The order is important. The healing ministry preceded the atonement, just as the prophecy of healing preceded the prophecy of Christ's atoning work. This means that particular healings cannot be a part of the atonement itself. The reason for the earthly healing work of Christ was "that ye may know that the Son of man hath power on earth to forgive sins" (Matthew 9:6).
The saving work of Christ includes deliverance from the death penalty for sin in one's past life, the power of sin in his present life, and the very presence of sin in the future life. Our great salvation has already delivered us from the eternal torments of the second death (Revelation 20:14, 21:8), from the defeating power of physical infirmities in our present bodies, and from the very presence of sickness and pain in the future. In any case, there is no question that God is well able in particular situations right now either to provide direct healing in answer to prayer, or sufficient grace to meet whatever physical need we have in a way that honors Him (2 Corinthians 12:7-9). HMM