I know that medical doctors affiliated with universities can teach at the medical schools as professors and write chapters in textbooks. But do medical doctors with a university affiliation also have to publish academic papers and write successful grant proposals, in order to be promoted?
Yes. Medical schools generate large proportions of the grant funding at many major research universities. Most faculty (except possibly research-only faculty) in medical schools have an M.D. degree. Research expectations for clinical-only faculty are often meager but faculty that conduct research at major universities often need to PI large grants to receive tenure.