Can you be fired for being out sick if you had a doctor’s note?

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