Does the delivery of a baby by a non-Muslim—whether male or female—affect the religion of the newborn and its Islamic فطرة (innate disposition), especially given that they do not pronounce God’s name or mention Him during their actions?

It is not permissible for a woman to expose her 'awrah (intimate parts) to someone unlawful for her, except out of necessity or severe need, such as childbirth. She must first seek a skilled Muslim female doctor. If none is available, then a skilled non-Muslim female doctor. If none is available, it is permissible to resort to a trustworthy Muslim male doctor. If none is available, there is no harm in a non-Muslim male doctor, provided her husband, a mahram (definition), or trustworthy women are present. Whoever goes to a non-female doctor initially, when one is available and without necessity, is committing a sin.

The delivery of a newborn by a non-Muslim or a fasiq (openly sinful) Muslim does not affect the child's religion or fitra (innate disposition). Rather, what affects their fitra is their upbringing and guidance after birth. "Every child is born upon the fitra, and it is his parents who make him a Jew, a Christian, or a Magian."