The first thing to remember when thinking about this counterfactual is that Christopher Columbus did not discover America, nor was he the first explorer to discover one of the American continents. Vikings had established settlements in Canada in the 900s, which were mainly used for supplying wood for ship building and repair. The Vikings attempted to establish a more permanent settlement, but the Native Americans proved to be too difficult to live near when the relationship between the Vikings and Native Americans turned hostile. There is also some evidence that explorers from Polynesia landed in South America around the same time that Columbus was establishing a foothold in the Caribbean. Some have even theorized that the Chinese had explored some of North America around 1300 BCE.
Columbus's exploration did not provide the first knowledge of the American continent, but it did give the European powers a better idea of the potential wealth of the continent, both in terms of natural resources and potential slaves. Columbus's four voyages provided a basis for conquest of South America by Spain. It may also have provided more incentive for the establishment of trading outposts by France, which likely motivated England to establish its own settlements. Without the Columbus voyages, America would most likely have been colonized at a later date, as reports of the continent existed and the European powers were in a race for resources. It is highly unlikely that a voyage would have been sponsored by one of these governments anyway.
Without Columbus's voyage, there may have been no Cortes voyage, or other later voyage by Spain. This would have prevented or delayed the destruction of the Aztec, Incan, and Mayan empires. It also would have delayed exposure of Native Americans to European diseases. If, instead of Columbus and later Spanish explorers, the initial European explorers followed smaller scale settlements, similar to the Vikings, then the rapid depopulation of North America may not have occurred. This could have significantly delayed the establishment of American colonies and would likely have made westward expansion far bloodier for Americans. If British colonies were established later it may have made it difficult for the colonists to obtain independence and, potentially, may have delayed or prevented the French Revolution as well.