Employer wants a doctor's note but I don't have one?

It is not an absurd policy. Many, many companies have that policy because of abusive employees in the past. You can't blame an employer - they need reliable help. Knowing the policy, you should have gone to the doctor. Failure to do so means you were not sick enough to stay away from work. Yes, you can be fired and, if you are, it is for cause.

How is it an "absurd" policy? It was designed to prevent people like you from taking sick days when you're NOT sick. You've lost your job because you're not responsible and apparently your job didn't meant that much to you. If you don't want to follow an employers company policy, create your own company with your own policies. What's that? You can't cause you're not capable of accomplishing such a task? Then you'd better get use to following rules.

If you weren't sick enough to need to go to the doctor, then you were well enough to go to work. Your employer has caught you out, admit your lies and hope he takes pity on you.

You took a sick day. You now must show that you are not too sick to work with food.

Go to the doctor, and get a note.

Or get another job.

You go to a walk in clinic.

You knew you needed a note....

I believe the law is that for the first five days of sickness you do not need a note, not every illness requires a doctor's visit, they are being stupid, and breaking employment law.

If your in the US contact DOL and ask them.
In Australia the Fair Work Ombudsman.

Well then, find a job with some regular humans. They can't survive with that policy.