There are many ways in which Ahab's obsession with Moby Dick can be understood as a kind of "cult," if we understand "cult" to mean a kind of secret, quasi-religious society that binds its members through oaths and ceremonies to the will of a charismatic leader. You can find many details that relate to this in chapter 36, "The Quarter Deck." For one, Ahab is clearly obsessed with the whale, "his one unsleeping, ever-pacing thought."
Ahab's meeting with the crew has the aspect of an enthusiastic revival meeting, in which Ahab is the charismatic preacher and the crew his congregation.
The doubloon Ahab nails to the mast is a kind of talisman or holy object; it represents both the hunt for the whale, and the reward for finding it. (The mystic significance of the doubloon is dwelt on in chapter 99, "The Doubloon.")
Ahab's admission that Moby Dick took his leg gives his obsession with the whale a new, personal character. The voyage has become one of personal vengeance, as the skeptic Starbuck notes—he is "game for his crooked jaw" if it "fairly comes in the way of the business we follow." In other words, he would kill Moby Dick like any other whale, but sees Ahab's thirst for vengeance as a kind of heresy, or, as he says, "blasphemous."
Ahab's famous explanation of reality as "paste-board masks" is a kind of alternate, heretical theology. It's clear that, for Ahab, Moby Dick is more than just a whale, and his killing it would be a kind of personal triumph over the divine.
Ahab's rhetoric convinces the crew of the righteousness of his cause. He has, in effect, brainwashed them. Even Starbuck has "inhaled" Ahab's enthusiasm.
Finally, The ceremony Ahab performs with the harpooneers to bind them to their pledge to hunt down Moby Dick is perhaps the best evidence of Ahab as a cult figure. The three mates act as "cupbearers" to the three harpooneers, who drink grog from the sockets (the "goblet end") of their harpoons, in an act that suggests a twisted, heretical kind of communion.