Why Doesn't moving a wire near earth generate electric current?

You are correct; moving a wire around earth would indeed produce a current in it. However, consider this one important fact. Earth is a 30 microtesla magnet. That's barely anything. It's the fact that we live on it that means we can detect it from anywhere. The current it would produce would be so low as to be negligible, and your wire's resistance would probably completely offset it.

Actually, moving a wire in the Earth’s magnetic field will generate a current. Due to the fact that the Earth’s magnetic field is so small, roughly 25 - 65 $mu$Tesla, (a fridge magnet has a magnetic field approximately 200 times stronger than Earth’s) the electric field generated in this manner would be so small that it would probably have no practical purpose.

In my freshman physics lab we had circular coils (about half a meter in diameter with a known number of turns) that could be flipped in a mounted frame. The output came from slip rings and was connected to a ballistic galvanometer. With the charge produced by a 180 degree flip, we could estimate the horizontal and vertical components of the earth's field within the lab.

It depends how you move the wire. The magnetic field of the earth is uniform in the small range of a lab. If the wire circuit is only translated, the magnetic flux doesn't change and there is no induced voltage.

If it is also rotated, there is a very small voltage, as written in the other answers. The intensity depends on the angular velocity.