By labeling Whitebeard's ship The Moby Dick, Oda makes this reference pull quadruple duty.
Firstly, it signals and foreshadows the Whitebeard crew's eventual destruction. Moby Dick is, fundamentally, a tragedy about a voyage the reader is fairly sure will fail from the start, in the pursuit of a goal which is fundamentally impossible (in this case, that fundamental goal stands in for both the crew finding the One Piece and Newgate himself trying to outrun his own mortality).
Second, it gives the uneasy feeling that Something Is Incorrect About Newgate, or at least the way he runs his ship, no matter what his "sons" say. Captain Ahab is a deeply destructive and sinister force on his own crew, and Melville makes it clear that God, Father, and Captain are all deeply entwined on the sea. He also argues that all on the "ship" should be wary of those forces, because they have the ability to warp and corrupt the fabric of someone's morality, ethics, and reality.
Thirdly, because Newgate is in many ways a symbol of the sea's old ways and bygone era in the early parts of the story, equating him with Moby Dick implies that the whole era's goals were somehow impossible or in vain. Whatever the Golden Age of Pirates was in this world, its goals were un-reached, the pursuit of those goals were deadly, and to go back to those ways would be chasing something that not only no longer exists, but never existed in the first place (this is fully paid off Marineford, which kills off not only Newgate, but Ace, effectively scrubbing the world clean from any active force that harkens back to Roger's days).
Fourth and lastly, and extended from point 3, Oda makes it clear that the Straw Hats, and with them their whole generation, aren't The Moby Dick. They operate their crews differently; they want different things; though they are chasing the same goal, the path to that goal is fundamentally different. Here is the much-discussed "New Era," who are all fundamentally not The Moby Dick, and in fact are often set against it. By making The Moby Dick a foil, the new generation seems more revolutionary, their methods seem more radical, and what they are chasing seems more actively possible in a way it was not for Whitebeard.
