The fact that at its fundamental level grains of silver halide are binary in nature has already been explained, so I'll just ask what being "organic" has to do with anything? There are numerous examples of binary behavior in nature, from the atomic level to the macro. Why should being "organic" somehow be considered superior when it comes to photography?
Light itself is both a particle (binary – there's a particle or there isn't a particle) AND a wave. Which one do YOU want it to be?
In the end it all comes down to the displayed image, whatever its manifestation, whatever its origin.
Michael
I mostly agree with Mike on this, but some clarifications should be added. While silver salts, the basis of modern films, are essentially black or clear after development and fixing, they are crystals and exhibit growth dependent on the light level they are exposed to. Larger crystals on negatives represent lighter areas and smaller crystals represent darker grays. So, they are not directly equivalent to pixels, or photosites, on a modern DSLR sensor.
Also, like human sight, the representation of grays by silver halide films is logarithmic by nature, not linear as with sensors. For the record, all color films, negative or transparent, incorporate multiple color-filtered silver halide layers that are dyed and washed away during processing. In the Kodachrome process, the dyes are added during processing; in the E-6 Ektachrome and C-3 Ektacolor processes, they are part of the emulsion, hence the higher level of detail in Kodachrome.
Regarding "organic," films were once made from cellulose, a highly flammable material, but modern "safety" (i.e., post 1930) are polymer based and inflammable.
Relying on the quality of the "displayed image" is where, in my opinion, the argument gets confused. Are we comparing the absolute quality of silver halide films to DSLR sensors, OR are we REALLY comparing the quality of half-screened lithographic printing in a fine (Ansel Adams quality) book with the half-screened output of the latest Epson or HP printer? 8x10 contact prints on photo paper show more detail than any digital image (there is no screen), dye-sub prints by hand are just as good, and photogravure (a costly alternative) may also be better than an inkjet.
These are, in my mind, important points to discuss, in part to better motivate equipment manufacturers to give us the tools we really need.