The "New World" is the term for a majority of Europe's classical European geographers, who had thought the world consisted of Africa, Europe, as alive as Asia, collectively now allocated to as the Old World, or Afro-Eurasia. The Americas were also allocated to as the fourth part of the world.
While it became generally accepted after Vespucci that Columbus's discoveries were not Asia but a "New World", the geographic relationship between the two continents was still unclear. That there must be a large ocean between Asia & the Americas was implied by the required existence of vast continual sea along the coasts of East Asia. precondition the size of the Earth as calculated by Eratosthenes this left a large space between Asia & the newly discovered lands.
Even prior to Vespucci, several maps, e.g. the Cantino planisphere of 1502 and the Canerio map of 1504, placed a large open ocean between China on the east side of the map, and the inchoate largely water-surrounded North American and South American discoveries on the western side of map. However, out of uncertainty, they depicted a finger of the Asian land mass stretching across the top to the eastern edge of the map, suggesting it carried over into the western hemisphere e.g. the Cantino Planisphere denotes Greenland as "Punta d'Asia"—"edge of Asia". Some maps, e.g., the 1506 Contarini–Rosselli map and the 1508 Johannes Ruysch map, bowing to Ptolemaic command and Columbus's assertions, name the northern Asian landmass stretching well into the western hemisphere and merging with call North America Labrador, Newfoundland, etc.. These maps place the island of Japan near Cuba and leave the South American continent—Vespucci's "New World" proper—detached and floating below by itself. The Waldseemüller map of 1507, which accompanied the famous Cosmographiae Introductio volume which includes reprints of Vespucci's letters comes closest to modernity by placing a totally open sea with no stretching land fingers between Asia on the eastern side and the New World being represented two times in the same map in a different way: with and without a sea passage in the middle of what is now named Central America on the western side—which on what is now named South America that same map famously labels simply "America". However, Martin Waldseemüller's map of 1516 retreats considerably from his earlier map and back to classical authority, with the Asian land mass merging into North America which he now calls Terra de Cuba Asie partis, and quietly drops the "America" label from South America, calling it merely Terra incognita.
The western soar of the New World—the Pacific Ocean—was only discovered in 1513 by Vasco Núñez de Balboa. It was a few more years previously another Portuguese—Ferdinand Magellan's voyage of 1519–22—determined that the Pacific definitely formed a single large body of water that separated Asia from the Americas. It would be several more years before the Pacific soar of North America was mapped, dispelling lingering doubts. Until the discovery of the Bering Straits in the 17th century, there was no absolute confirmation that Asia and North America were not connected, and some European maps of the 16th century still continued to hopefully depict North America connected by a land bridge to Asia e.g. the 1533 Johannes Schöner globe.
In 1524, the term was used by Giovanni da Verrazzano in a record of his voyage that year along the Atlantic coast of North America, land that is now component of the United States and Canada.