Using a GAN discriminator as a standalone classifier
Yes, we can use the Discriminator of the GAN to classify images. But we should make sure that the images produced by the Generator are real looking. 


If you have trained your GAN on a large number of images and it is performing pretty well on the dataset then I insist you to treat the Discriminator model as a pretrained model ( like we do in transfer learning ) and again train this model on images which were not used to train the GAN earlier. Thus the model is fine tuned on the dataset on the GAN wasn't trained before.
Another similar way could be to only use the weights of the CNN layers and load them in our new model.



  Suppose I train a GAN until the generator is good enough to fool the
  discriminator much of the time. Could I then use the discriminator as
  a classifier that tests whether an input belongs to a single class?


You should definitely try reusing a Discriminator from a GAN and share your results too :-).