Artificial intelligence (AI) describes systems capable of making decisions of high cognitive complexity; autonomous AI systems in healthcare are AI systems that make clinical decisions without human oversight. Such rigorously validated medical diagnostic AI systems hold great promise for improving access to care, increasing accuracy, and lowering cost, while enabling specialist physicians to provide the greatest value by managing and treating patients whose outcomes can be improved. Ensuring that autonomous AI provides these benefits requires evaluation of the autonomous AI's effect on patient outcome, design, validation, data usage, and accountability, from a bioethics and accountability perspective. We performed a literature review of bioethical principles for AI, and derived evaluation rules for autonomous AI, grounded in bioethical principles. The rules include patient outcome, validation, reference standard, design, data usage, and accountability for medical liability. Application of the rules explains successful US Food and Drug Administration (FDA) de novo authorization of an example, the first autonomous point-of-care diabetic retinopathy examination de novo authorized by the FDA, after a preregistered clinical trial. Physicians need to become competent in understanding the potential risks and benefits of autonomous AI, and understand its design, safety, efficacy and equity, validation, and liability, as well as how its data were obtained. The autonomous AI evaluation rules introduced here can help physicians understand limitations and risks as well as the potential benefits of autonomous AI for their patients.