It seems not in the first place, because suffering pertains to persons, not natures. Now Christ is one person in two natures, consequently if the person of Christ suffers, then the union of the two natures suffers. Ergo, the divine nature as well as the human one suffered.
In the second place, whatever is not assumed is not redeemed. If God does not suffer, he does not redeem suffering. But we must assert most strongly that God redeems the suffering of the world through Christ, therefore God must have assumed the suffering of the world.
On the contrary, is the opinion of the council of Rome:
If someone were to say that in the passion of the cross God felt pain and not the flesh with the soul, which he had put on–the form of a slave, which he himself had taken, as the Scriptures say: he does not think correctly.
I respond, that it is important to distinguish the sense in which a human is a person and the way in which God is a Trinity of persons. In the human case you have one substance (This particular concrete instantiation of the form ‘humanity’) and therefore you have one person (Socrates). In the case of the Trinity you have one substance (deity) and three persons (the Father, the Son and Holy Ghost). In the case of Christ you have two substances (one human nature and one divine nature) which are united hypostatically in one person (the Son).
Socrates is passible because he is a mutable contingent being. The Father is clearly impassible because he is necessary, eternal and therefore unchanging. But does the Son suffer? It would seem not, because what one member of the Trinity does, they all do, since they all share one substance. Since the Father does not suffer, therefore the Son cannot suffer.
On the other hand, it would seem the the Son must suffer because the Son is truly human and contingency, mutability and therefore passibility pertain to the human nature of the Son just as they pertain to the human nature of Socrates.
But clearly the Son cannot both suffer and not suffer in the same respect, therefore, I answer that the Son suffers in his human nature and does not suffer in his divine nature. It is true to say “Christ suffers” because Christ’s human nature suffers. It is also true to say, “In Christ God suffers” because Christ is God and Christ suffers in his human nature. But it is not true to say “The Father suffers” because the Son suffers only in his human nature and not in his divine nature, therefore his suffering is not an action held in common with the other persons of the Trinity. Therefore the opinion of the council of Rome is entirely correct.
In response to the first objection that only persons suffer, therefore both natures suffer, I note that the person of Christ is fundamentally different than our ordinary experience of human persons insofar as he is truly divine as well as truly human. In the case of Socrates it is clear that his human nature suffers when Socrates suffers, but one cannot extrapolate from this that when Jesus suffers therefore his divine nature as well as his human nature suffers.
In response to the second objection, that God must assume suffering to redeem it, it suffices simply to note that evil is nothing positive. God assumes human nature, but suffering, as an evil, is a defect or privation in human nature, not a positive feature of it. God assumes all creaturely reality in order to redeem it, but he does not assume evil, precisely because evil is not part of creaturely reality except as a privation.