What characteristic of Romantic literature does Herman Melville's "Moby-Dick" contain?

The "Dark Romanticism" in Moby Dick is also closely related to the love of the sublime that English and American writers from the late-1700s to the early-1800s relished (primarily the Gothic writers from 1790 to 1830).  The meaning of "sublime" in today's vernacular is quite different from the way "sublime" was used several centuries ago.  Then, sublime primarily had this meaning: "impressing the mind with a sense of grandeur or power;inspiring awe, veneration, etc." (Dictionary.com).  Things that were "sublime" were absolutely unexplainable, but literature of that time period attempted to explain the unexplainable, and, in falling short, further evoke a sublime reaction in readers.

The end of Chapter 16 is a perfect example of Melville's use of the sublime.  At this point Ishmael and the reader have learned only a little about Captain Ahab.  The narrator's curiosity about him piques the reader's curiosity as well:

As I walked away, I was full of thoughtfullness; what had been incidentally revealed to me of Captain Ahab, filled me with a certain wild vagueness of painfulness concerning him.  And somehow, at the time, I felt a sympathy adn a sorrow for him, but for I don't know what, unless it was the cruel loss of his leg.  And yet I also felt a strange awe of him; but that sort of awe, which I cannot at all describe, was not exactly awer: I don not know what it was.  But I felt it; and it did not disincline me towards him; though I felt impatience at what seemed like mystery in him, so imperfectly as he was known to me then. (This is from p 79 of the 2nd Norton Critical Edition)

Ishmael is mysteriously attracted to, while simultaneously repelled, by Ahab, and he has no idea why--the feelings are beyond his control.

Once you start to notice Melville's use of the sublime, you'll see it everywhere--in his characterizations of Queequeg, in the way he describes the sea, and certainly in Ahab's relationship to Moby Dick.

What makes Moby Dick a piece of romantic literature?

Romanticism was a literary movement in which authors and artists reacted to changes that occurred during the Industrial Revolution. During the Industrial Revolution, America became a hotspot of factory-led activity, and with it, population growth. Cities became overrun with people, and without modern advancements such as indoor plumbing and trash collection, they also became dirty and grimy.  Romanticism was a reaction to this time period of industrial growth.

Romantics cherished nature above all else, and many of them were pulled to the wilderness of America. The wilderness was untouched by the current advancements and provided a reprieve from the dirt and overpopulation. Romantics believed in the self over society and valued intuition and emotion over logic and reason; they also believed in the importance of dreams, symbols, and the power of nature over mankind. Romantic texts usually had a romantic hero - someone who had rejected society's norms and whose life revolved around only himself.

Thinking of these characteristics, Moby Dick most definitely qualifies as a romantic text. Captain Ahab, the captain of the ill-fated ship, the Pequod, is driven by his own emotions in pursuing the giant while whale, Moby Dick. Logically, it's completely unreasonable to think he can locate one whale in the entire ocean, but he continues anyway.  He puts a ship full of men in mortal danger in order to pursue his over vengeance. To him, Moby Dick is the embodiment of evil - again, an illogical idea. Captain Ahab is a Romantic hero - he has rejected his societal role and taken up this obsession with the white whale. Finally, nature (the whale) is successful in the end, triumphing over man.