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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.
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I traveled the roads to Chaco Canyon last week. My vehicle was a BMW F650 GS (adventure type motorcycle). We entered the park from the south via Hwy 57. The road was in pretty good shape, note that it was dry. We passed several cars going over the road with no problems. The surface was firm, you could see deep tracks on the lower areas, suggesting the road could be interesting when muddy.
We left by the north route (7950). It was very wash boarded and rough for the first several miles. The road was certainly worse than 57. We passed several cars taking it slow, this may not be the most comfortable way to approach wash board. The road becomes asphalt at 7900 and is good to Nageezi, which is where we exited. There is a gas station at Nageezi.
