Can my employer fire me while on medical leave if I have a doctor's note requesting the leave?

If you actually took Family and Medical Leave Act leave (see below for criteria for doing so), then you may not be fired for taking the leave--though you could be fired for an unrelated reason (such as violating company policy, a restructuring, etc.). If you did not actually take and use FMLA leave, however, the employer may fire you even for being out on medical leave unless the employer itself specifically authorized it--your doctor has no authority over your employer, and the employer is not required to follow the doctor's instructions. And you could also be fired for unrelated reasons.

FMLA leave is available if your company qualifies (which means it has at least 50 employees within a 75-mile radius) and you also qualify (you have worked there more-or-less full time for a year); also, the leave must be for a sufficiently serious condition, which generally means one requiring either 3 days or more in-patient stay, or repeated doctor-visit follow-ups. (Go to the Department of Labor website to see the exact criteria.)