Does the Coriolis Effect affect a wind from East to West?

The full answer is that you're moving through the Ekman layer. In the upper atmosphere, away from the ground, the wind direction is such that the Coriolis force and the pressure gradient balance each other, for the most part. Near the surface, you also have turbulent friction/drag and all three components of the force have to balance. If you follow the math on that Wikipedia page and work through a few examples, you should be able to convince yourself that the wind turns by about 30 degrees clockwise as you go from the surface to the free atmosphere in the northern hemisphere, regardless of the wind direction in the free atmosphere or the height of the boundary layer.

Observations show evidence of the Ekman spiral, though it's definitely not as clean as the theory and the number of 30 degrees is an approximation. I'm actually more familiar with the Ekman spiral from oceanography, where the key results are that the water on the surface of the ocean flows 45 degrees clockwise from the wind, and the net boundary layer transport of water is 90 degrees from the wind.

First forget about the cardinal directions and accept the next notions about the directions of forces relative to the air flow. The air parcel is affected by the pressure force towards the low pressure. The Coriolis force pulls air 90 degrees to the right relative to the current wind direction. The friction force that affects the flow is always directed to the opposite from the current wind direction. Assuming that the flow is in equilibrium (it is approximately most of the time) the next schematic applies (in the Northern Hemisphere) (from weather.gov training page).

The left part of the schematic applies when there is no friction (i.e. not near the surface) and the wind is parallel to the isobars therefore called Geostrophic wind. The right part is with friction where the wind direction has turned approximately 30 degrees counterclockwise relative to the case aloft.

I deduce from your question that you are also grasping on why the Coriolis force is to the right of related to the wind direction. In fact there is no Coriolis force, but an effect that is seen as an apparent force due to our perception fixed on the rotating earth as the frame of reference. Basically the air parcel only appears to be forced to the right/left in the Northern/Southern Hemisphere. This is quite hard to explain intuitively and I remember first learning this through videos and animations like this open course video of rotating reference frames.

You all got it wrong! Coriolis force has nothing to do with the centrifugal force.

The main answer to the question why Coriolis doesn't apply in the East-West direction is that East-West direction is not a straight line, so they are already curved themselves.

No object on Earth can travel dead East or dead West when shot from a source if there is a constant turn applied to it The parallels of the Earth are not straight lines, only the equator is.

The Coriolis Effect only has to do with geometry and actually doesn't even care about North, South, East or West: it works by the same amount in all directions. You can take a pencil out and see how it works purely geometrically.

Also the "vertical" component of the Coriolis effect is there (Eotvos effect), but it is not needed to explain this question nor to explain the formation of depressions as it is so small that it plays no part in this.