Saw a video that mentioned something I hadn't heard before--he's not just against employers/government having to pay for birth control, he's also against having to pay for a prenatal test known as amniocentesis, where a large needle is inserted into the placenta in order to collect a sample and do a direct DNA test on the baby. Pretty much the only reason this is ever done is to confirm a suspected genetic disability the baby may carry, and the procedure itself carries a risk of causing a miscarriage. So since the only reason one would really need to know this information before the child's birth is if one were planning to abort the baby if a severe disability is confirmed, he wants to disallow it.
On the other hand, at least he's not a hypocrite: he has one child with
Trisomy 18, and another baby that had something else wrong that only lived 2 hours after being born.