I don't know how many thousands of students passed through the gates of the Federal Law Enforcement Training Center while I was there, but we saw many folks in every class who were cross-eye dominant, so I can say with confidence that it makes no difference at all when shooting a handgun. It does, however, when shooting a long gun.
Col. Jeff Cooper, who started Gunsite, told me one time that he was cross eye dominant, but when posing for photos pretended to be right eye dominant.
The problem is when one is neither left nor right eye dominant. Years ago, they taught folks to just close one eye, but...under great stress, especially when one is startled, it is not possible. For that reason when we encountered a student with that problem we issued a set of shooting glasses with a little Scotch tape on one lens of his/her shooting glasses.
If they were right handed, the tape would be on the left lens and if they were left handed it would be on the right lens. It doesn't take the brain long to "compensate" for blurred vision in one eye by making the other dominant. (It speeds up the process if the shooter wears the training glasses for short periods other than just at the range.)
Hope that helps,
-kent