The Sharingan allegedly mutated from the Byakugan (although this seemed to be retconned) so it's probable that one of the eyes are dominant and the other recessive. And if this is the case, then because the Uchiha clan broke off from the Hyūga, then it's probably that the Byakugan is dominant, and thus their child would be born with the Byakugan, not the Sharingan. Further support is that the Byakugan affects the phenotype from birth (white eye) whereas the Sharingan is only activated for a select few.