He did, several times. Infact Arthur coughed and touched alot people before even chapter 6. It's actually a miracle and unrealistic no one else in the camp got sick with it. He shared drinks, food and tobbaco with people, got his TB bacteria over everything he touched and coughed on. I'm going to say the main reason would be, would you rather be able to riddle Micah with bullets, or have him slowly die off screen?
Also, tuberculosis is a sickness where only few of the infected get sick and actually have latent tuberculosis. So Micah could have had TB and never even noticed (and also not infected no one). Additionally, how could Arthur have known that coughing on people infects them? Remember, this takes place 125 years ago, and Arthur is not exactly an educated guy ("German?! I can barely speak English").
