Is the angular resolution of a telescope irrespective of used eye-piece? Suppose a telescope has aperture $D = 20$ cm. The angular resolution of such telescope, according to the Rayleigh criterion (if I have understood it correctly), is given by 

$$\theta = 1.22\cdot\frac{\lambda}{D} \text{ rad}$$

where $\lambda$ is the wavelength of light and $D$ is the aperture. Suppose $\lambda = 550$ nm (mid-point of visible light). Then 

$$\theta = 1.22\cdot\frac{\lambda}{D} \text{ rad}
= 1.22\cdot\frac{550 \text{ nm}}{20\text{ cm}} \text{ rad}
= 1.22\cdot\frac{5.5\cdot10^{-5}\text{ cm}}{20\text{ cm}}\cdot 206264.8''
\approx 0.69''
$$

In other words, the angular resolution of the telescope is $0.69$ seconds of arc for $\lambda=550$ nm.

Now, is it really so that the eye-piece of the telescope has no play in this at all? Wikipedia says that


  and D is the diameter of the lens' aperture.


which I would understand to be taken as the size of the primary mirror alone in a Newtonian telescope.

So if there is for example a double (binary) star, I will see them as one no matter what kind of magnification/eye-piece I am using if the angle between the stars (from my point of view) is less than $0.69''$? And vice versa, if the angle between the stars is greater than $0.69''$ I will see them as separate stars with any eye-piece or magnification?

(In theory, I know there's a lot of limitations from optics, seeing, etc.)
The resolution of a telescope is the resolution of the image created by the primary mirror at the focal plane.  It provides the minimum separation between two equal brightness stars that appear separate in that image.  Often, astronomers put a photographic plate or a CCD at the focal plane, create long exposure photos and these can then be examined at arbitrary resolution (with a magnifying glass, microscope, or zoom software on a screen).  But, no matter how much you magnify the image, stars separated by less than the resolution will appear merged.  

It is true that if you use, say, a wide-angle lens when observing directly with a telescope the resolution that you get can be limited by the resolution of your eye.  What is happening is the telescope is forming an image on your retina that has the full resolution, but your eye cannot make use of the finest details.  On the other hand, a hawk looking through the same eyepiece may get full resolution benefits since hawks' eyes have better resolution than humans'.