I do not think that Christ could have sinned because of two reasons:
1. Even though he had two natures - human and divine - he was but one person; and that person is the Second Person of the Blessed Trinity. Whatever Christ did was the action of a divine person, so just as it is correct to say that God died (on the Cross); it is impossible to say that God should turn away from himself in sin.
2. Even within the human nature (human intellect, will, and body) of Christ, it was not possible for Him to sin, if one holds that He enjoyed the beatific vision from the moment of His conception. Allusions to Him having the beatific vision include passages wherein he speaks of knowing the Father, of seeing the Father, etc. So, just as the saints in heaven cannot sin - because seeing God there is no other good greater to them, which could entice them, or even tempt them to choose the inferior good, so in Christ, He could not be mistaken, or ticked into choosing a contradictory lesser good. We can sin because things opposed to God may appear to be superior goods at the particular time of temptation - but they only seem superior, because we do not see God and His infinite goodness.
These two modes of impeccability are classically termed 1. Physical impeccability; and 2. Moral impeccability. (The term "physical" was interchangeable with "existence-based" until modern times).