Can my employer fire me for being out sick even if I had a doctor’s note?

In most states, a worker has no legal protection if they call out sick, whether or not they have a doctor's note. The laws that protect an employee due to an illness only afford coverage if they have a serious medical condition or disability. Further, most employment is what is known as "at will" which means that a company can set the conditions of work much as it sees fit. Accordingly, an employee can be fired for any reason or 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 an 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, they are entitled to up to 12 weeks of leave for a serious medical condition. Additionally 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 will explain further.
Bottom line, your employer's action was legal unless it violated the terms of any an employment contract or union agreement, or it constituted some form of legally actionable discrimination.