In the Northern Hemisphere, since you're travelling towards the East due to the Earth's rotation and don't change latitude, when you drop an object from a large height, the object will continue to travel in a path independent of the Earth's rotation (although the initial velocity is in the direction of the Earth's rotation) and as such it'll follow a great circle path, which is basically going the longest way around the Earth, encircling the entire globe. So yes, it will go into the Southern Hemisphere and to the person that dropped the object, it'll look as though the object deviates towards the South (to the right if you're looking East).
I don't know what the OP means by "it'll fall in a curved spiral". That's just not true, and clearly shows his lack of understanding of physics.