What if God abandoned God?
Well if you mean "Divinity abandoned Divinity", that would be silly, as natures can't abandon something.
If you mean God [the Father] abandoned God [the Father], what does that have to do with the Son?
What Christ is questioning is not the existence, nor the power, nor the ability, but the seeming silence of the One addressed as "My God." As Raymond Brown said, this unrelenting silence, Christ is so isolated and estranged that He no longer uses "Father" language by speaks as the humblest servant.
This is the first time ever that Christ does not address God by "Father" but as "My God". But all his cry brings out is the silence of God. The remaining conclusion to that psalm is that He did not really feel abandoned, however we must interpret that saying of Christ in His situation and not of the psalmist. It needs to be read in light of the life of Christ, but that's not to deny the prophecy either.
As you question, Is the Father separated from the Son? Is their communion broken, even if it be for a single moment? These aren't easy questions but they have to be answered because at its core, it is who Christ really is.
Two questions then, why was Christ been abandoned by the Father? And how would it be possible for such an abandonment to take place?
The first question is at the fall of Man. Man, because of sin, is in a state of separation from God, or better is abandoned
by Him. As St. Cyril said "We had become accursed through Adam's transgression and had fallen into the trap of death, abandoned
by God." (source
) Note here this isn't neglect from God but is on the sin of man, for it was man that caused the abandonment to happen, however God has not forgotten.
Like the quote I posted of St. Gregory, whatever has not been united to the limitless, infinite, perfect, divine Son of God, cannot be made perfect, infinite and limitless. This is indicative of the energies of God that are communicated through the Incarnation. If Christ did not take to Himself flesh and become a real man, then humanity could not be saved. Things like the mind, body, emotions, etc could not be healed had Christ not assumed.
So in order to bring man back into union with God, to overcome that abandonment, Christ would need to take upon Himself the abandonment of humanity.
And again according to St. Cyril, the limitless Son of God would need to be indescribably become the abandoned; He who was united to the Father from before all ages would need to become forsaken so that He could undo our abandonment by His obedience and complete submission. For only then coujld our abandonment be assumed, sanctified, and restored into life united with God. St. Cyril is quick to point out that this cry from Christ of abandonment, came from humanity which was abandoned by His divinity should be seen as madness and blasphemy. However, as I'll address the second question, the divinity of Christ is an inseparable union with His humanity. Again from St. Cyril, he says that every act involves the soul and body together. Now before the OO's on this board lash at me for trying to separate two natures, however they come together to constitue a new condition by their union without compromising the integrity of either nature. But by uniting the to His humanity, in the Person of Christ, He may experience abandonment in His flesh, but not as God nor merely as a man but as the one God-enfleshed person of Christ.
So as God, the Son cannot possibly be abandoned, for He is in perfect union with the Father, however by uniting to His humanity, in the person of Christ, the son may experience abandonment in His flesh (just like Christ can feel pain, suffering, anguish, tears, etc). Eventhough God cannot be abandoned, God-enfleshed can be. The Son of God who became man was literally abandoned on the cross by the One He called Father. Abandoned by man, nature and God. He felt within Him the severing of an indissoulouble bond. This was an exclamation of an incomprehensible truth. But this wasn't the end here, because what was not possible before the Incarnation, became possible after it. By assuming the human condition, by becoming one of the abandoned, by enduring loneliness and anguish and by tasting seperation from God in a way that we can never imagine nor describe, Christ assumed, overcame and sanctified our
abandonment. He brought together an impossible chasm, bringing man once more into unity with God. That cry on the cross was like an explosion of suffering in love. What may seem as the indescribable division in the life of God is actually the delcaration of unity. In this moment of abandoment Christ identifies Himself with the will of His Forsaker, and in doing so reveals the love of God for man, the very revelation of the Father HImself. By emptying Himself, he gives humanity a glimpse into the intimate life of God.
It isn't a cry of despair.
And I'm done being an pseudo armchair theologian for tonight. I know the word "God-man" rubs people the wrong way on this board, so maybe God-enfleshed suffices, or maybe it doesn't. But you are extremely bright and I'm sure you understand what I'm saying. Waiting for Severian, Salpy or mina's rebuttal to my seperation of natures.
tl;dr version is, man was abandoned from God, and for Christ to restore the unity with God, he had to assume abandonment. In a sense to heal it.