You can seek a medical discharge under Army Regulation 40–501, which I shall post the relevant section below.
However, let me provide you with information on the process as I have shared with other seeking medical discharge.
Medical Disability Status
Schedule a visit with the military doctor, who is legally entitled to start the process of getting a medical discharge. A complete medical evaluation will be performed. The results will then be processed through the disability evaluation system that is set in place to review each request.
Realize that some conditions, such as a missing limb, is an obvious qualifier for initiating the process to qualify for a military medical discharge. Other disease processes are not so obvious and sometimes require initiation by a civilian doctor.
Evaluate any condition in relation to the safety and well-being of the service member and the other members. If the medical condition makes the member unfit to perform her duties without imposing unreasonable requirements on other members, she is a candidate for a military medical discharge.
Be on active duty and entitled to basic pay when the medical condition was incurred in order to get the benefits of a medical military discharge.
Provide a civilian doctor with the appropriate documents outlining the specific information required to initiate a request for medical disability. This may include past medical records to substantiate the claim of present injury or exacerbation of an injury or illness that prohibits performance of required duties.
Realize the service member cannot apply for disability discharge. The request must be made by either a medical officer or the service member's commanding officer with a full explanation of the illness.
Consider the Medical Evaluation Board the first step. Once the Board determines there is enough evidence to support a consideration of medical discharge, the case is referred to the Physical Evaluation Board.
Have your evidence reviewed by the Physical Evaluation Board, which is responsible for assessing all documented evidence to determine fitness to remain in the military. If the determination is for discharge, the percentage of benefits for disability will also be determined
Army Regulation 40–501
2–23. Lungs, chest wall, pleura, and mediastinum
-d. Asthma (493), including reactive airway disease, exercise-induced bronchospasm or asthmatic bronchitis, reliably
diagnosed and symptomatic after the 13th birthday, is disqualifying. Reliable diagnostic criteria may include any of the
following elements: substantiated history of cough, wheeze, chest tightness, and/or dyspnea that persists or recurs over
a prolonged period of time, generally more than 12 months.
e. Current bronchitis (490), acute or chronic, symptoms over 3 months occurring at least twice a year (491), is
f. Current or history of bronchiectasis (494) is disqualifying.