What resolution and or DPI would a TV or VR headset need to be to equal 20/20 vision? and 20/10? With even the 4K TV it does not seem as sharp as my own vision.
To measure acuity, it's important to have a measure that considers visual space rather than physical world space (it doesn't make any sense to talk about "millimeters" because that says nothing about how far away the object is), so one would normally talk in terms of "degrees or arc" where 360 degrees is a whole field in all directions. Within degrees of arc, you can refer to "arcminutes" which are 1/60th of a degree of arc. If you could see everything in front of you, but nothing behind you, then you have 180 degrees of visual field in both horizontal and vertical dimensions (humans actually have a much smaller visual field than this), or 10,800 minutes of arc.
Your visual acuity greatly depends on where on the retina an image falls; the 'fovea' is the center of the retina where your vision is most sensitive. In the fovea, the maximum acuity is somewhere around 0.5-1.0 minutes of arc (see for example Rossi, E. A., & Roorda, A. (2010). The relationship between visual resolution and cone spacing in the human fovea. Nature neuroscience, 13(2), 156-157.), which corresponds nearly to cone density in the fovea.
A good "rule of thumb" (no pun intended, since we won't be using the thumb here...) is that the width of your fist at arms length is about 10 degrees of arc, your pinky finger is about 1 degree of arc. To get 0.5 arcminute resolution, then, you would need a screen the size of your fist (600 arcminutes) held at arms length to be 1200 pixels wide. Modern cellphones held at a moderate distance likely come near this limit, whereas TVs and monitors typically do not; however, you probably view your phone much closer than arms length and your TV much further.
Realistically, however, pixel counts far below this are sufficient for most scenes, because acuity measured this way is referring to the ability to separate points of light. In real natural vision, we aren't usually separating points of light, but rather organizing visual representations into salient objects, so the actual pixel density required to get a smooth image is less. Pixel limitations are more likely to be apparent when displaying text, since contrast is much higher with text than with natural scenes.
Rossi, E. A., & Roorda, A. (2010). The relationship between visual resolution and cone spacing in the human fovea. Nature neuroscience, 13(2), 156-157.