Round Rock is a Navajo community and census-designated place in Apache County, Arizona, United States. The population was 789 at the 2010 census. It is named after Round Rock, a nearby mesa. According to the United States Census Bureau, the CDP has a total area of 14.4 square miles, of which 14.2 square miles is land and 0.12 square miles, or 0.…Round Rock is a Navajo community and census-designated place in Apache County, Arizona, United States. The population was 789 at the 2010 census. It is named after Round Rock, a nearby mesa. According to the United States Census Bureau, the CDP has a total area of 14.4 square miles, of which 14.2 square miles is land and 0.12 square miles, or 0.68%, is water.