I forget where I read this explanation, but I'll share it, because I once had the same question about Christ's human nature: Could Jesus have sinned? In short, no. The answer lies in the fact that He was always, beyond doubt, in full and perfect communion and unity with His Father. Anyone who maintains that level of communion with the Father cannot sin. I'd dare say that we've had saints that come very close to perfect communion with God, but even they will fall short of the level of union Christ has with His Father, because they were of one essence. The issue isn't whether or not Christ had a fallen nature: He was born, fully human into a fallen world, just as we all are. But because He was perfectly united at all times with the Father, sin could never touch Him. That's my 2 cents.