The question is almost laughable. The US is so culturally different from Saudi Arabia. The US has already been there and done that, regarding lots of Saudi policies, and while sometimes there is slight regression, on the whole we continue to move forward. At the heart, Saudi Arabia and other countries like it are theocracies, and that is one thing the US will never be.
I believe it is more likely Saudi Arabia will evolve into a culture similar to the U.S. than the U.S. will return to an era where women couldn't vote or run for elective office. Also, Saudi Arabia has punishments for some crimes that would be considered draconian by U.S. standards. I do not see the U.S. overhauling its penal code to be comparable to that in Saudi Arabia.
Even if the U.S. made the decision to drill all the oil we had, we would never compare to Saudi Arabia. Or if I could say, Saudi America could never compare to the U.S. While the U.S. has an imperfect government, it is a much more fair, equitable, and progressive government than Saudi Arabia could have in the foreseeable future.
The idea that the United States, a secular constitutional republic, is somehow slated to become the next Saudi Arabia, a theocracy operating on the judgments of the Qu'ran, is ridiculous. The only parallel that could be slightly be made is that the United States is now producing more oil than it once did.