That's a very good question. Why had God forsaken His Son?
(In what sense had God forsaken His Son is another very good question, but for this post I'll use "forsaken" in the sense that Jesus meant when He spoke these words on the cross.)
Could we say that abandonment by the Father is the ultimate proof of the Incarnation?
The Incarnation is the act by which God became man. It took the Church centuries to fully articulate what had been revealed about this, and I doubt a single generation has passed since then without a lot of Christians who didn't really believe it. Mondays, Wednesdays, and Fridays, Jesus is God in a man-suit; Tuesdays, Thursdays, and Saturdays, He is a man with a Divine pacemaker next to His heart.
But that Friday, He was the God-man forsaken by God. Only man can be forsaken by God; the angelic spirits have already chosen their sides, and non-spiritual creatures don't have the kind of relationship with God that can admit of forsakenness.
Jesus was a man, like us in all things but sin, and the "all things but sin" includes the capacity to be forsaken by His Father, a capacity He does not have as the Eternal Son.
So when He was forsaken by His God, Jesus didn't disappear, or evaporate, or deflate like a balloon. He cried out, and He died.