I am meant to start my new job on May 2nd, but I got a terrible back injury. I'm in a lot of pain and unable to move around a lot or think clearly. I might need 1 week extra to recover. The company were pretty flexible with the start date.
Whether it is unprofessional is irrelevant. In your situation, you don't have a choice other than delaying your start date. If it takes a week for your back to recover, then that's how long it takes.
Mail the company requesting to delay your start date because of your back injury. Then call your manager or HR to discuss the situation. It may also be a good idea to add some buffer, so that if your recovery takes longer than a week, you don't have to delay the start date again. Requesting them to let you join sooner than agreed (if you recover early) would be definitely less embarrassing.