Can an employee be lawfully terminated after missing days of work due to a newly arisen medical problem?

The answer is:  it really depends on other factors in addition to the illness.  Generally speaking, an employer can fire or terminante the employment relationship for attendance reasons or no reason... because Texas is an at-wtill employment state.  Some employees do, however, have a bit more protection than others in the form of FMLA (Family Medical Leave Act).  FMLA essentially provides that if an employee files for FMLA leave and qualifies, then an employer must hold their employment open while they are on leave.  FMLA does not apply to all employers and all employees.  For example, only employers with more than 50 employees are covered.  So if the sick employee works in a real estate office with only 10 employees, then they would not be entitled protection under FMLA.  FMLA also does not cover part-time workers. In order to get the protection of FMLA, the employee must request the leave and complete the required paperwork.  If an employee doesn't ask for the FMLA leave, they can't later complain about a termination based on attendance issues related to the condition.  Even though an employee may not qualify for FMLA, they still may have the right to file for unemployment-- the disqualification of one relief does not automatically disqualify the employee from other types of relief.