Can an employer fire you if you are under doctor's care?

Yes.  However, if your employer is covered by the Family and Medical Leave Act and you meet the eligibility requirements, the employer would be required to allow you to have that time off. In general, the employer must have 50 or more employees with in 75 miles of the location where you work and you must have completed 12 months of service and accumulated 1250 hours in the 12 months leading up to your leave.  There are a couple of other requirements as well, but those are the big ones.