Can an astronaut standing on the moon observe earth's rotation with the naked eye?

Yes, as long as one was patient.  If one was on the Moon and stared up at the Earth and Washington DC was in the center of their view, in about thirty minutes, Lexington, Kentucky will have taken its place and in less than an hour Chicago, Illinois will have taken its place.  Add another thirty minutes and he or she would be looking at Topeka, Kansas.