Q:

Does water have a minimal thickness because of van der waals force?

I have read this question:
The thickness of a puddled sheet of water
https://en.wikipedia.org/wiki/Surface_tension#Puddles_on_a_surface

The liquid mass flattens out because that brings as much of the mercury to as low a level as possible, but the surface tension, at the same time, is acting to reduce the total surface area. The result of the compromise is a puddle of a nearly fixed thickness.
    The same surface tension demonstration can be done with water, lime water or even saline, but only on a surface made of a substance to which water does not adhere. Wax is such a substance. Water poured onto a smooth, flat, horizontal wax surface, say a waxed sheet of glass, will behave similarly to the mercury poured onto glass.
Another way to view surface tension is in terms of energy. A molecule in contact with a neighbor is in a lower state of energy than if it were alone (not in contact with a neighbor). The interior molecules have as many neighbors as they can possibly have, but the boundary molecules are missing neighbors (compared to interior molecules) and therefore have a higher energy. For the liquid to minimize its energy state, the number of higher energy boundary molecules must be minimized. The minimized number of boundary molecules results in a minimal surface area.

Now i do understand that in this more QM answer, it says that the molecules at the boundary must be minimized (because they have higher energy). This minimizes the size (and thickness because of gravity).
What this does not explain is, though, why this minimalization does not continue until only a single layer molecules are left on the surface.
This minimalization must stop somewhere where the forces equilize, that is:

gravity
van der waals

https://en.wikipedia.org/wiki/Van_der_Waals_force
https://pubs.acs.org/doi/10.1021/jp806376e
Am I correct that it is the van der waals force that keeps the molecules in a droplet and that gravity is trying to flatten the droplet as much as possible, thereby thinning the water, making as little number of layers of water molecules on top of each other as possible?
Question:

Is it the van der waals force that keeps the minimal size of the water droplet and the minimal thickness of the water (that is the minimal number of layers of molecules)?

A:

What this does not explain is, though, why this minimalization does not continue until only a single layer molecules are left on the surface.

Because the force of surface tension is trying to minimize the surface, not the thickness of water, since surface tension acts like a rubber sheet covering it and trying to pull it all together.
In case of the contact angle being $\theta = 180^°$, when the surface doesn't attract water at all, is the shape only a result of gravity and surface tension. The thickness would be minimized only if there were no surface tension (because then there would have to be no compromise between minimizing thickness (gravity) and minimizing surface and therefore making the liquid into a sphere (surface tension)).
The contact angle depends both on the liquid and the surface, so in order to have liquid of volume $V$ completely covering surface $A$ with a layer as thin as possible $\left(\approx\frac{V}A\right)$, we'd need such a combination of liquid and surface for which $\theta = 0^°$. An example of such a combination are ordinary surfaces under normal temperature and pressure, and water with dissolved carbon (and other components) that usually covers ordinary surfaces with a liquid layer several nanometers thick.

Am I correct that it is the van der Waals force that keeps the molecules in a droplet and that gravity is trying to flatten the droplet as much as possible, thereby thinning the water, making as little number of layers of water molecules on top of each other as possible?

Yes, but in addition to van der Waals force and gravity, there are also hydrogen bonds that also try to keep molecules in the droplet.

Is it the van der Waals force that keeps the minimal size of the water droplet and the minimal thickness of the water

Van der Waals force and hydrogen bonds only try to keep minimal surface (i.e. as spherical as possible). It's only gravity that tries to keep the minimal thickness of the water (as long as we neglect the attraction between the liquid and the solid surface it lies upon, otherwise there would be a fourth force).

