If and how are the Coriolis force and gravitation related? I recently got into a discussion whether and how the Coriolis force/effect is related to gravitation.
Is gravitation involved in the Coriolis force? If so, how could that be explained?
 A: They really aren't related.  The Coriolis force is essentially conservation of momentum.  Remember that the Earth is (roughly) 24,000 miles in circumference at the equator, and so everything "at rest" on the equator is moving eastward at 1,000 mph.  Away from the equator the ground's eastward speed is slower, roughly 
$$
v_\text{east} = v_\text{equator} \cos \theta
$$
where $\theta$ is the latitude.
So if you have an air mass that moves from the equator to 45º north latitude, it has an "extra" 300 mph eastward speed compared to the ground; if you have air move from 45º north to the equator, the ground at the equator is outpacing it by 300 mph; the net effect is that any amount of north-south motion gets mixed up with east-west motion and gives you swirly storms.
The only role played by gravity is that gravity keeps us all stapled to the earth's surface.
A: Corirolis force is simply due to a moving frame of reference.  If you were on a merry-go-round, and threw a ball, it would go straight in the fair-ground frame of reference, but because you see it go away from the direction you threw it, you might suppose it experienced a force.  
The earth spins, and because we don't suspect this, we attrit normal inertial motion to a force.  
Gravity just keeps us on the ground, and has nothing to do with bending balls in fairgrounds.
