Board Thread:Naruto Discussions/@comment-25442710-20140920121351/@comment-1512913-20140920130223

I don't think a Sharingan is a prerequisite at all. Hagoromo's chakra is both Indra and Ashura's chakras and Indra's chakra contains the Sharingan. So having Hagoromo's chakra should give you a doujutsu even without prior possession of one.

Also Obito giving Kakashi his chakra made him manifest Obito's Sharingan, so having Hagoromo's chakra should make one manifest his Rinnegan. The only logical explanation therefore is that Hagoromo gave Naruto only more Ashura chakra or so, or in case he gave him the full deal, then Naruto's Ashura's chakra hasn't mixed yet with the Indra part, but that doesn't explain why it did in case of Sasuke.