If a prevailing wind traveled due east in the Northern Hemisphere, would the Coriolis Effect act on the water?
The Coriolis Effect occurs in all directions equally, even straight zonal (east-west) winds.  

Many indeed struggle with this idea - this Physics SE question has a fair explanation as to why.

As arkaia commented, the total deflection magnitude is $f=2{\omega}\cdot{\sin(latitude)}$, which is an orthogonal vector change (and as such shows up as terms in both the zonal and meridional velocity equations).

So there's still deflection to the right (in the NH) equally regardless of which direction the air/water is flowing (except that in such a feature, deeper water is moving slower, so deflects by less magnitude... but still matching in regards to the proportion in comparison to the velocity).  Otherwise the result would not be a spiral... but only at most a 90° decaying curve, since as the water begins to turn towards a more zonal direction, the angle of deflection would start to decrease, until at some height the water is moving easterly (or westerly) (such as at height labeled 3 in this diagram), and the water below it wouldn't turn further.  So the direction independence of Coriolis is vital to there being Ekman spirals at all.