Can we prove the Earth rotation with a disk mounted in its center on a frictionless axis?

If the disk is spinning then it will act as a gyroscope and will try to keep its axis oriented in a constant direction in space. This is called a Foucault gyroscope and it can be used to demonstrate the rotation of the earth, although great care must be taken to minimise friction and allow the axis of the gyroscope to rotate freely. See this Wikipedia article for more details.

What you describe is reminiscent of a 1913 experiment by Arthur Compton. This setup is referred to as a Compton ring

A circular tube is filled with water (with suspended particles in the water to allow tracking of motion of the water).

The initial position of the tube is perpendicular to the local level surface. The water in the tube is allowed to come to complete rest. This rest state is a state of co-rotating with the Earth rotation. Then the tube is flipped 180 degrees. After that flip the water is seen to have been set in motion, the magnitude of the velocity can be observed with a microscope.

This setup will show the strongest effect at the Equator, and a smaller effect on higher latitudes.

So, contrary to assertions in comments and answer to this question using gyroscopic effect is not the only way to demonstrate the Earth's rotation. However, if a disk is used that is initially co-rotating with the Earth then the setup does need to execute a flip in order to obtain any data