Is it rude when someone doesn't return an E-Mail like this?

In the office I work in emailed excuses are not allowed so if I were you I would look up the company handbook and make sure that you didn't commit the transgression. And an email like this should be followed-up upon by YOU. Call your supervisor if you can and check to make sure they got the message. Also keep your doctor's note because you may need a copy to prove that you actually needed the time off, the Human Resources officer will want to put it in your file.