Generally when people are talking about the road "through Monument Valley" they are talking about the loop within the Navajo Tribal Park. That road is a dirt road that is fairly rough. While many cars could make it without too many problems, driving it with a rental would likely be a violation of the rental contract. There are guided tours through the loop that are available for those who don't want to drive.

That said, Monument Valley is not limited to the tribal park, and there are plenty of great views on other roads in the area, including US-163.