Nathaniel Hawthorne wrote "The Birthmark" at a time when the scientific method was being glorified and people were starting to think science really could take us anywhere we wanted to go. He set his story about 60 years earlier, in the 100-year-long wake of the Newtonian Revolution, in the Age of Enlightenment, when science was gaining its momentum. His story argues that, despite the general optimism, science really does have its limitations. There are certain things that humans are not privileged to know, not capable of doing. It is not only ignorant, the story seems to say, but downright dangerous to try and play God.
Hawthorne assails Positivism in his story.
Hawthorne does not oppose science in his story; he merely posits that science has its limits.