It is true that the rotation of the Earth gives rise to an inertial force that will be experienced by liquids and gases on the Earth. This is called the Coriolis effect and it accounts for why cyclones rotate clockwise in the southern hemisphere and anticlockwise in the northern hemisphere.
The Coriolis effect is dominant in large quantities of gas and liquid under the right conditions. However, in a sink or bath there are many other more dominant factors that will affect which way the water goes down the plughole. The shape of the container, the direction in which the water was added, the movement of the water molecules and any outside disturbances all have an impact.
There have been many studies trying to eliminate the dominance of these other factors and test whether the Coriolis effect has any bearing whatsoever on the direction in which water chooses to rotate down a plughole, but it is too weak a force on a small body of liquid to have any noticeable effect.
Rik Sargent, Science Museum