If you had a doctor’s note, can you be fired for being out sick?

 In moststates/cities, you have no legal protection if you call out sick. This is true whether or not you have a doctor's note. The laws that protect an employee due to illness afford coverage only if they have a serious medical condition or disability. The fact is that most employment is "at will". This means that a company can set the terms and conditions of work much as it sees fit. Therefore, an employee can be fired for any reason or no reason at all, and this includes calling out sick.
That having been said, under the Family Medical Leave Act (FMLA), if a company has 50 or more employees within a 75 mile radius of an employee's workplace, if they've worked at least a year, and if they have worked at least 1,250 hours in the past year, they are entitled to up to 12 weeks of leave for a serious medical condition (note: the term "serious" is limited)). Also, a worker may be covered under the Americans with Disabilities Act (ADA), so if they are out sick for a disability (including medical treatments, surgery, etc.), they may have protection. The Department of Labor's website explains further.
Accordingly, your employer's action was legal unless it violated the terms of any applicable a union agreement or employment contract, or constituted some form of legally actionable discrimination.

Yes, you may be fired for being out sick, even with a doctor's note, unless:
1) You had and properly used some form of paid time off (PTO), like sick days, to cover the absence; or
2) Your company was covered by, you were eligible for, and the absence was one you could use FMLA leave for, and you properly used FMLA leave.
Other than as the above, the law does not make employers keep employees who have unexecused (by the employer) absences from work; and the doctor has no power to excuse you, since he/she is not a manager, executive, or owner of the company/employer.