A raster image is basically a grid of squares and in order for you to have something that looks like a dot or diamond, the dot size must be larger than the grid size. Obviously from your question, a pixel is not round enough so to describe the roundness properly, you are probably going to need at least 3x3 pixels (which looks like a cross).
So in order to pack a 3x3 "circle" into the same "optical space" you need more pixels in order to simulate roundness in a square world.
The next step is deciding the size of the simulated dot screen, measured in lines per inch. FYI newspapers are/were approximately 80 lpi. Commercial color is about 150 lpi.
To translate this into photoshop, convert your image to greyscale. Then convert it to bitmap. For the output resolution, try 1200ppi. For the halftone frequency, try 80 (smaller numbers yield a larger pattern).
The image gets upsampled to your output resolution and then converted to a bitmap simulating halftone. This will increase the file size: a greyscale pixel is stored optimally as 1 byte, so a 1000 px square is 1,000,000 bytes (about .9MB). A bitmap pixel is optimally stored as 1 bit (1/8 the storage of a greyscale pixel, but we are upsampling the image by some amount. If you are starting with a 1000px square at 72 dpi and request upsampling to 1200dpi, you are increasing the dimensions by nearly 17x in each dimension (=289000000 bits = approx 34MB)
TLDNR; need moar pixels.