The use of the generic term `God` is misleading. Jesus`s Spirit was leaving Him. The HolySpirit is His Spirit and He was leaving the body that He had formed. The HolySpirit is the strength and life of the man Jesus.
So, we see a separation of the Word from the Spirit which had never happened and will never happen again. They are one, Jesus being the IncarnateWord and the HolySpirit being His Spirit.
Death is foreign to the HolySpirit who is life. As Jesus for the first time and only time was separated from the HolySpirit He cried out in anguish.
In the vacuum created by His HolySpirit leaving Him, the sins of the world entered His body and soul. The anguish that He felt was two-fold: -> His HolySpirit left -> All sin was poured into Him.
Even in His last moments Jesus fulfilled the Word. Psa 22:1-* is what He was referencing in His last cry.