To understand how a camera's pixel pitch may affect Depth of Field (DoF), you must first understand what DoF is as well as what it isn't.

Regardless of the aperture of a lens, there will only be one distance that will be in focus. That is, there will only be one distance at which a point source of light will be focused to a single point on the recording medium. Point sources of light at other distances will be projected on the sensor (or film) plane as a blur circle, or circle of confusion (CoC). If this CoC is sufficiently small enough to be perceived as a point by human vision at a specific display size and distance, it is said to be within the DoF. The limits of DoF change based on aperture, focal length, and focus distance as well as the display size and viewing distance of the image. You can print two copies of the same image file and if one is displayed at twice the size of the other at the same viewing distance by a person with the same visual acuity the smaller print will appear to have more DoF than the larger one (assuming the resolution of the image file itself is not the limiting factor). There is no magical barrier at which everything on one side is in perfect focus and everything outside of that line is blurred. Rather, as the distance from the true point-of-focus increases, so does the size of the blur circle and we gradually begin to perceive that objects are not absolutely sharp.

Your fundamental understanding of how an image is created out of the data from a sensor with a Bayer mask in front of it (the overwhelming majority of digital cameras) is not entirely accurate. There is no direct correlation to a single sensel (pixel well) on a Bayer sensor and a single pixel in an image produced from the data provided by that sensor. The numbers for each color of each pixel in the image produced are interpolated from the data produced by multiple adjacent sensels.

You also assume, incorrectly in most use cases, that a two pixel blur will be detectable at typical viewing sizes and distances. It won't be. A typical circle of confusion for FF cameras is 0.03mm (30µm). A typical 20MP FF sensor has pixels around 6.5µm wide. Even accounting for the 2x2 grid of RGGB masked pixels used to produce 4 RGB pixels in the image produced, the 13µm width of a 2x2 cell is still less than half the width of the 0.03mm CoC needed for an 8x10 print viewed at 10-12 inches by a person with 20/20 vision. Most APS-C cameras have pixels that are slightly larger than 4µm wide. So the recommended CoC for an APS-C sensor of around 0.019mm is still over twice as wide as a 2x2 cells on a typical APS-C sensor.

If the theoretical sensor in your question with pixels 4X larger are sufficiently large enough to limit the perceived resolution of the image, then everything in the image with a CoC smaller than the resolution limits of the sensor will appear to be equally in focus at the expense of also appearing equally pixelated/blurry. This would occur when the CoC needed for a particular display size and viewing distance is less than twice the width of the pixel pitch of the camera's sensor. It would not, however, be a hard limit but instead would the point at which we gradually begin to perceive that the picture is made up of individual pixels that our eyes can discriminate.