What makes this story so authentically "gothic" is the fact that Charlotte Perkins Gilman herself suffered from nervous problems leading to melancholia; her prescribed cure was also bed rest. For, the horror comes directly from the mind of the author--the psychological torment of the narrator is truly horrific.
As with Poe's narrators, the reader, cannot be certain whether the telling of the story is realistic or simply the outpourings of a disturbed mind. The atmosphere of "The Yellow Wallpaper" is pervaded with this sense of mystery, hopeless isolation, fear, preternatural sensations. The victim of her husband's isolating her either because of the doctor's orders or because of his own domineering desires senses fear in a terribly oppressive manner. Her very being is tormented.
Gothic fiction generally features the kinds of things you find in good horror movies. Fear, the supernatural, secret rooms and isolated houses, suspense and dread are some of the ingredients gothic writers use to make their stories effective. Gilman makes us of several of these ingredients. The story takes place in an isolated estate which is in a state of disrepair. The narrator is kept in a room all by herself with strange, yellow wallpaper. At first, she has a feeling of dread which turns to hallucination and progresses to full blown insanity. She keeps thinking there is a a ghostly woman trapped inside the wallpaper. The woman's isolation is reinforced by the fact that no one will take her seriously. These are classic characteristics of the gothic genre.
The American Gothic movement sprang from Individualism. The Individualists focused on the ideals of mankind, and on trusting yourself and your instincts. Their intense focus on the individual nature of mankind eventually led to the Gothic movement, which focused on individuals also, but their dark side, their potential for evil. The Gothic movement emphasized the power of the mind to create horror, fear, and reveal our darkest selves. Mankind was no longer ideal and holy, but instead capable of weakness, insanity, and vice.
In "The Yellow Wallpaper", the narrator is dealing with her slowly crumbling mental health. It focuses on her delusions and paranoias, and we see that the human mind can create a horrifying reality, and lead to unimaginable distress. From the setting (an abandoned and creepy house), to the isolation (she's trapped in a room with horrid wallpaper, all by herself), to the bizarre state of her mind, there are Gothic themes throughout.