What aspects of Romanticism are explored in "The Birthmark"?
This is an interesting question because it gets to the heart of one of the main themes in "The Birthmark."
One of the most important and pervasive elements of Romanticism is a preference for and respect of the natural world over the world of science. Other goals of Romanticism include the celebration of natural beauty, innocence, and emotion.
In "The Birthmark," there are three characters--Aylmer, the scientist; Georgiana, his wife; and Aminadab, Aylmer's assistant. Aylmer, because his entire world is encompassed by science and experimentation, represents the anti-Romantic, primarily because he believes that he has control over Nature; his wife, Georgiana, represents the natural person--beautiful, loving, with a trusting heart, and innocent; Aminadab, who is described almost as an elemental creature who looks like he just popped out of the ground, represents natural man, but at a base level.
Aylmer, who believes only in the power of science to control the natural world, conceives a completely irrational hatred of a minor blemish on Georgiana's cheek, which is shaped like a tiny hand (perhaps the hand of Nature). What's worse is that Aylmer convinces Georgiana, who trusts Aylmer impicitly, that the hand, the "blemish," is so horrible that it has to be removed. Aylmer, the supremely confident scientist, believes he can "correct" Nature's mistake.
The natural world asserts itself even in the character of Aminadab who, when he realizes that Aylmer intends to remove the birthmark, instinctively knows that trying to alter Nature's work is not going to succeed, and he concludes by saying that he would be happy to have such a beautiful woman as his wife.
Of course, Aylmer's attempts end in disaster for Georgiana but not before we learn that Aylmer has created a poison that allows him to determine when someone is going to die, Hawthorne's explicit criticism of man as God. In a conflict between science and Nature, especially in works based on the precepts of Romanticism, science is not going to conquer Nature.