The easiest way I can find to do it would be to get your background shape & then convert it to a bitmap. Add a gaussian blur. Then convert the bitmap to a black & white halftone. You can change the shape of the screen & increase the number of dots per line to fit your needs. Once that is done, you can to an autotrace or just leave it as a bitmap.
The correct method can be accomplished in Photoshop.
When you converted it to a 1-bit bitmap, did you change the settings all the way down to 1 dot per inch? If you did & that is still not large enough, maybe making the image smaller or reducing the resolution would in essence make the dots larger when you convert it.
old post sorry, have any of you tried this program it works a treat. It converts any jpeg to vector tones (dot size etc can be achieved with a little playing about). You can use the on line version or download it. I found it years ago.
http://homokaasu.org/rasterbator/wizard.gas?Phase=1