I am attempting to design a web icon, which is required to be 120 x 30 at 72 dpi JPEG or PNG in it's final state. I designed in Photoshop at 1200 x 300 72 dpi. When I try to export as PNG/ JPEG, it look like total blurry garbage, or if I try to utilize the resample options, it ups the dpi. Is there no way to export a small web image that is clear when viewed at actual size?
You could lose the text and focus on making the actual logo the icon/ button. As Kevin already pointed out it is unreasonable to expect more details at such low resolutions and trying to retain the text is questionable to begin with.
Agree with Kevin: do this at actual pixel size. Use the pencil tool, not the brush tool, to work at precise pixel level.
Pixel art isn't easy! You can't just scale down a bigger file.
Oh, and the ppi number doesn't matter on screen, disregard that. It could be 1 ppi or 489 ppi and it wouldn't make any difference. If there is a "requirement" for a certain ppi, someone doesn't understand what ppi is. What you need to consider is the pixel size, nothing else.