Your humble question: Why is asthma sputum white?
My humble answer: Asthma is a condition that often results in an increase in the number of goblet cells that line your lungs. Goblet cells are cells that produce sputum. During an exacerbation of asthma these cells increase production of sputum.
It's important to know your lungs are normally sterile. It's normal for your body uses sputum to trap particles that you inhale to prevent them from getting to your lungs and to keep the lungs germ free. This sputum makes it's way to your oral airway for you to swallow or spit up. Usually this sputum is so scant you won't even notice it, and if you do notice it it's clear or white.
If you are sick with a lung infection your sputum produciton may increase as a natural mechanism to rid your otherwise sterile lungs of the infecting microbes. Depending on the infecting agent, this sputum will come in a variety of colors. To learn what colors represent what condition check out the sputum lexicon.
Asthma sputum is unique in that it is not produced to fight off an infection. Actually, the ironic thing about asthma is it's a disease whereby your immune system works too well. It's your body thinking something you inhaled that is normally considered safe (like dust mites for example) is an enemy bacteria. Or it's your immune system attacking your own body.
For this reason, the increased sputum will not contain bacteria, and it will not contain viruses. It will be sterile. In other words, it was not produced to kill off an infection (that is, unless an infecting agent triggered the attack). For this reason, asthma sputum is sterile. It is white.