Can you install an air dryer on a 1 gallon air compressor?

You're missing the principle about a compressor in this and your other question, a compressor brings in air from the outside, the cylinder(s) compress the air and in the process create heat, warm air carries more moisture but since it is mostly just ambient humidity it won't draw much more. What happens in the tank is the compressed air from the cylinder(s) expands rapidly and releases energy (heat) and the moisture will condense. So, you can't prevent moisture from entering the tank, but you can certainly filter from the tank to your tools/equipment. After every use, you need to open the tank petcock to drain out the condensed water. It's the condensation of the air coming in that causes rust in a tank.

They make liquid gases in a similar manner, the air is compressed and released into heat radiator tanks to remove heat, not add cold, then that air is compressed and re-released into expansion tanks to remove more heat till the component gases liquefy at different points. We used compressed air drills underground mining and after a while, the exhaust manifolds would actually frost over because of the volumes of air being released, and the ambient temp was near 100 degrees F, so it also would make the heading cooler when we drilled. 

The dryers are for outgoing air so you are not getting water in your tools, especially bad when painting.

it wouldnt be practical, but you could install an in-line dryer to filter moisture if your painting for example .. that would be simple ..