Question: "What is the understanding of the Church on Matthew 27:46? I have done some research myself and I have seen everything from the idea that Christ was abandoned by the Father, to arguments that Christ was not abandoned, nor was He distressed, but was proclaiming that in this darkest hour, so to speak, the Father is still with Him."
We find the text you reference in Matthew, as well as in Mark 15:34, but it is also clear that Christ is quoting from Psalm 21:1 (which is Psalm 22:1 in Protestant Bibles). The entirety of Psalm 21[22] is seen as a prophecy of the death and Resurrection of Christ by the Fathers, as is either clearly suggested or made explicit in the crucifixion accounts in all four Gospels. Psalm 21[22]:16-17 ("they have pierced my hands and my feet. They have numbered all my bones, and they themselves have looked and stared upon me") which is alluded to in John 19:37, and Psalm 21[22]:18 ("They have parted my garments amongst themselves, and for my vesture have they cast lots") is directly quoted by Matthew 27:35 and John 19:23-24, and clearly alluded to in Mark 15:24 and Luke 23:34.
And so to find the answer to this question we need to see what the Fathers say about these passages.
St. Gregory the Theologian (329-390) emphasizes that these words are spoken by Christ on our behalf, because He suffered on our behalf, but that there was no separation between the Father and the Son, and that Christ's humanity was never separated from His divinity:
"It was not He who was forsaken either by the Father, or by His own Godhead, as some have thought, as if It were afraid of the Passion, and therefore withdrew Itself from Him in His Sufferings (for who compelled Him either to be born on earth at all, or to be lifted up on the Cross?) But as I said, He was in His own Person representing us. For we were the forsaken and despised before, but now by the Sufferings of Him Who could not suffer, we were taken up and saved. Similarly, He makes His own our folly and our transgressions; and says what follows in the Psalm, for it is very evident that the Twenty-first Psalm refers to Christ.
The same consideration applies to another passage, “He learnt obedience by the things which He suffered,” and to His “strong crying and tears,” and His “Entreaties,” and His “being heard,” and His” Reverence,” all of which He wonderfully wrought out, like a drama whose plot was devised on our behalf. For in His character of the Word He was neither obedient nor disobedient. For such expressions belong to servants, and inferiors, and the one applies to the better sort of them, while the other belongs to those who deserve punishment. But, in the character of the Form of a Servant, He condescends to His fellow servants, nay, to His servants, and takes upon Him a strange form, bearing all me and mine in Himself, that in Himself He may exhaust the bad, as fire does wax, or as the sun does the mists of earth; and that I may partake of His nature by the blending. Thus He honours obedience by His action, and proves it experimentally by His Passion. For to possess the disposition is not enough, just as it would not be enough for us, unless we also proved it by our acts; for action is the proof of disposition.
And perhaps it would not be wrong to assume this also, that by the art of His love for man He gauges our obedience, and measures all by comparison with His own Sufferings, so that He may know our condition by His own, and how much is demanded of us, and how much we yield, taking into the account, along with our environment, our weakness also" (Fourth Theological Oration 5-6).
St. John Chrysostom (347-407) adds that Christ, by citing this Old Testament prophecy, bore witness to the Old Testament, and showed that He was not in opposition to it, but that it bore witness to Him:
"And for this reason, even after this He speaks, that they might learn that He was still alive, and that He Himself did this, and that they might become by this also more gentle, and He saith, “Eli, Eli, lama sabachthani?” that unto His last breath they might see that He honors His Father, and is no adversary of God. Wherefore also He uttered a certain cry from the prophet, even to His last hour bearing witness to the Old Testament, and not simply a cry from the prophet, but also in Hebrew, so as to be plain and intelligible to them, and by all things He shows how He is of one mind with Him that begat Him" (Homily 88 on the Gospel of Matthew).
St. Jerome (347-420) in his commentary on Matthew, emphasizes the fact that Psalm 21 is clearly about Christ and no one else, and that the humility of the words cited point us to the scandal of the Cross (which St. Paul speaks of in 1 Corinthians 1:18-2:5):
"He has used the beginning of the twenty-first Psalm. Moreover, he leaves out what is read in the middle of the first verse: "Look upon me." For the Hebrew it reads: "My God, my God, why have you forsaken me?" Therefore, they are impious who think that this Psalm was spoken under the persona of David, or of Esther and Mordecai. For the evangelists understood the testimonies taken from it of the Savior, as for example: "They divided my garments among themselves and cast lots for my clothing"; and elsewhere: "They have pierced my hands and my feet." Do not marvel at the humility of the words and the complaint of the forsaken one. For by knowing the "form of a servant," you see the scandal of the cross" (The Fathers of the Church: St. Jerome, Commentary on Matthew, 4:27:46, trans. Thomas P. Scheck (Washington, D.C.: Catholic University of America Press, 2008), p. 319).
Blessed Theophylact (1050-1107) expands on St. John Chrysostom's commentary, and then adds some additional insights:
"Jesus speaks prophetically in the Hebrew tongue to show that He does not contend with the Old Testament. He said, "Why hast Thou forsaken Me?" to show that He was truly man, and not just in appearance. For man avidly desires life and has a physical appetite for it. Just as Christ agonized and was sorely troubled before the cross, showing the fear that is ours by nature, so now He says, "Why hast Thou forsaken Me?" displaying our natural thirst for life. For He was truly man and like us in all respects, but without sins. Some have understood it in this manner: the Savior spoke on behalf of the Jews and said, "Why hast Thou forsaken the Jewish race, O Father, that it should commit such a sin and be handed over to destruction?" For as Christ was one of the Jews, He said "forsaken Me," meaning, "Why hast Thou forsaken My kinsmen, My people, that they should bring such a great evil upon themselves?" (The Explanation of the Holy Gospel According to Matthew. Fr. Christopher Stade, Trans. (House Springs, MO: Chrysostom Press, 1992), p. 247f).
Blessed Theodoret (393-458), in his commentary on Psalm 21[22] emphasizes the prophetic focus of the Psalm as a whole:
"This Psalm foretells the events of Christ the Lord's Passion and Resurrection, the calling of the nations and the salvation of the world" (The Fathers of the Church: Theodoret of Cyrus, Commentary on the Psalms, 1-72, trans. Robert C. Hill (Washington, D.C.: Catholic University of America Press, 2000), p. 145).
And on the specific verse in question, he writes very much along the lines of St. Gregory the Theologian:
"Now, it was while fixed to the wood that the Lord uttered this cry, using the very language of the Hebrews, "Eli, Eli, lema sachthani?" So how could the testimony of truth itself be found inadmissible? He says he has been abandoned, however, since, despite no sin having been committed by him, death prevailed after receiving authority against sinners. So he calls abandonment not any separation from the divinity to which he was united, as some suspected, but the permission given for the Passion: the divinity was present to the form of a slave in his suffering and permitted him to suffer so as to procure salvation for the whole of nature. Of course, it was not affected by suffering from that source: how could the impassible nature suffer? It is Christ the Lord as a man, on the contrary, who speaks these words..." (The Fathers of the Church: Theodoret of Cyrus, Commentary on the Psalms, 1-72, trans. Robert C. Hill (Washington, D.C.: Catholic University of America Press, 2000), p. 146).
Cassiodorus (485-585), in his commentary on Psalm 21[22] echoes the other Fathers, and cites St. Cyril of Alexandria to reinforce them:
"He asks the Father why he has been abandoned by Him. These and similar expressions seek to express His humanity, but we must not believe that divinity was absent to Him even at the passion, since the apostle says: If they had known, thy would never have crucified the Lord of glory [1 Corinthians 2:8]. Though He was impassible, He suffered through the humanity which He assumed, and which could suffer. He was immortal, but He died; He never dies, but He rose again. On this topic Father Cyril expressed this beautiful thought: Through the grace of God He tasted death for all, surrendering His body though by nature He was life and the resurrection of the dead" [Cyril of Alexandria, Ep. 17 (MG 77.113B)]... He broadcasts the experiences of the humanity which He assumed, repelling words of blasphemy and impious mouthings, for He says that words begotten by sins are far from Him. The salvation of His sacred soul was not to embrace the speech of sinners, but gladly to endure by the virtue of patience what He suffered through God's dispensation" (Cassiodorus: Explanation of the Psalms, Vol. 1, trans. P. G. Walsh, (New York: Paulist Press,1990), p. 217).
So in summary, Christ Himself, by quoting the first words of Psalm 21 from the Cross, pointed us to the words of this prophetic Psalm, so that we would understand the meaning of His death, which He suffered for our sakes and in our place. There is no sense in which Christ was separated from the Father while on the Cross, but He voluntarily suffered the abandonment of the penalty for our sins in His humanity. And though this Psalm begins with words that speak of abandonment, and speak of cruel suffering, they end with words that speak of Christ's resurrection, his victory over death, and the salvation of the Church, which would be drawn from all nations:
"I will declare Thy name unto my brethren, in the midst of the church will I hymn Thee. Ye that fear the Lord, praise Him; all ye that are of the seed of Jacob, glorify Him; let all fear Him that are of the seed of Israel. For He hath not set at naught nor abhorred the supplications of the pauper, nor hath He turned His face from me; and when I cried unto Him, He hearkened unto me. From Thee is my praise; in the great church will I confess Thee; my vows will I pay before them that fear Thee. The poor shall eat and be filled, and they that seek the Lord shall praise Him; their hearts shall live for ever and ever. All the ends of the earth shall remember and shall turn unto the Lord, and all the kindreds of the nations shall worship before Him. For the kingdom is the Lord's and He Himself is sovereign of the nations. All they that be fat upon the earth have eaten and worshipped; all they that go down into the earth shall fall down before Him. Yea, my soul liveth for Him, and my seed shall serve Him. The generation that cometh shall be told of the Lord, and they shall proclaim His righteousness to a people that shall be born, which the Lord hath made" (Psalm 21:22-31).