Can someone avoid being fired if they are under a doctor’s care?

Not necessarily.

IF the condition for which care is being sought is so serious, so impactful in the individual's life, that it would qualify as a disability under the law (i.e. it's something which materially prevents or limits the person's ability to do some of the normal functions of life), then the employer cannot fire him or her because of the condition if (1) the employee has brought it to the employer's attention and asked for an accomodation; and (2) some reasonable accomodation, or alteration in work duties, conditions, or tools, will allow the person to do the job. The accomodation must be reasonable in that it cannot be too costly and must let the person do a job which the employer needs done.

Note that a person who is disabled can still be fired for other reasons: unexcused absenteeism, violating employer policy, poor performance not related to the disability, etc.

Also, if someone is taking protected leave of some kind, like Family and Medical Leave Act Leave, to deal with a condition, they cannot be fired for using the leave; though again, they can be fired for unrelatede reasons.

So being under a doctor's care does not prohibit termination on other grounds; and even vis-a-vis medical issues, like disability, the protection, while strong, is narrow in scope.