Will this move due to pressure difference? A U-tube is placed on a frictionless horizontal plane with a cork fixed tightly at the middle of the horizontal arm. It is filled with water and mercury on either side, as shown in the picture.

Let's consider points A and B (marked as red dots). Since A is facing mercury which has a higher density, there pressure is higher than that of B. So there is a net force acting on the cork from A to B. As the cork is connected with the U tube tightly, will the U tube accelerate towards the right?
 A: Remember that Pascal's law tells us that a fluid exerts its pressure equally in all directions, so if we draw all the forces that the mercury exerts on the container is in they will look something like this:

So while you are quite correct that the mercury is exerting a force to the right on the cork, it is also exerting a force to the left on the opposite side of the container. To find the net force you have to add up all the forces by integrating them, and when you do this you will find the net force the mercury exerts on the container is zero. This means the pressure of the mercury cannot make the container move.
This also applies to the water that we put in the right side. Indeed it applies for any fluids or indeed if we leave one side empty. Whatever combination of fluids we choose the net force on the container remains zero and the container does not move.
If the cork were free to slide then as the cork slid to the right the container would move to the left, but that's another question.
A: 
As the cork is connected with the U tube tightly, will the U tube accelerate towards the right?

The operative word is tightly implying the cork cannot move.
This means that as there are no external horizontal forces acting on the system (U-tube and liquids)  and as the vertical external forces are balanced, the system will not move.
A: The forces are internal to the U-tube system, and so will not cause movement. If they did, we'd have the basis for perpetual motion.
A: NO
It will not move.
Just imagine the  same thing without water column. Theres just a L shaped tube filled with mercury.
It won't  move u know.  Beacuse itnernal forces always maintain equilibrium and hence dont cause motion even if surface is frictionless
