I'm wondering if Naruto could have awakened the Sharingan or the Byakugan, since he was the reincarnation of Ashura Otsutsuki and Ashura's brother was of the Uchiha clan, and Ashura's father the Sage of Six Paths, who had a brother who was of the Hyuga which means Naruto is related to the Uchiha clan, the Senju clan, and the Hyuga clan.
No, Naruto could not unlock neither the Sharingan nor the Byakugan. It doesn't matter whether he was a reincarnation of Ashura. As a matter of fact, Ashura himself didn't possess the Sharingan or Byakugan, despite the fact that his predecessors had different dojutsu. So, it is absolutely clear that Naruto can't use the Sharingan or Byakugan unless he transplants them from someone.