The name United States of America was proposed by Thomas Paine  and was used officially for the first time in the Declaration of Independence, adopted on July 4, 1776. It is usually said in abbreviated form United States. Sometimes it is incorrectly called the United States of North America, resulting in a confusion in its name.
In Spanish, the use of North America as an abbreviated form of the name of this country is not acceptable, since there are other nations that share the North American subcontinent. Similarly, America should not be used to refer exclusively to the United States, even though it is a very widespread custom among English speakers to use the name of the continent as an abbreviated form of the name of the nation. When writing, the abbreviation USA is usually used (obligatorily with intermediate space and points as it is an abbreviation and not an acronym) and, to a lesser extent, the acronym USA. In Spanish it is totally incorrect, although frequent, the use of the English acronym USA.
At a time when Spain was a colonial power, was called American to anyone who had been born in some part of their dominions in America  , in the UK, an after century of discovery, called americans settlers who they lived in the portions of North America that this other kingdom was occupying, and that included the annexed territory called Nova Francia.
The influence of the United States in Europe and in the world has practically contributed to monopolizing the name for itself. The use of “American” (in American Spanish) is now very widespread in English-speaking countries and in other languages due to its influence, although in the Spanish language the most common name continues to be that of Americans.
It should be taken into account that the term ” American ” is not exclusive to people who live in the United States, American is anyone who lives in America, taking into account that America is a continent.
The United States emerged from the British colonization of America, led by waves of British immigrants who, between the seventeenth and eighteenth centuries, founded Thirteen Colonies on the Atlantic coast of the North American subcontinent, east of the Appalachians. These colonies turned their backs on the French possessions of Québec and Louisiana.
After a rather peaceful development of the settlers, the wars against the French to the north forced the creation of colonial army corps, one of the first expressions of national identity. Later came uprisings such as the Boston Harbor Tea Party(1773). The repressive measures of the English government provoked the beginning of the War of Independence.  The colonists formed an army of militiamen that were placed under the command of George Washington, who had problems equipping his men with weapons and ammunition, in addition to not having a fleet to fight that of the British Empire, so asked for help from France, which to retaliate from the Seven Years’ War agreed to help the colonies.
The initial development was clearly under English rule, but its course would change when after the Battle of Saratoga, the first great American victory,  France and later Spain entered the war supporting the North American independentistas.
In 1783 by the Peace of Versailles, England is forced to recognize the independence of the Thirteen British Colonies, as they had written in the famous Declaration of Independence of the United States of 1776. 
Independence and expansion
According to oxfordastronomy.com, the official date of the founding of the United States is the 4 of July of 1776,  when the Second Continental Congress, representing the Thirteen Colonies British secessionists, signed the Declaration of Independence. However, the structure of government underwent a great change in 1788 when the Articles of Confederation were replaced by the Constitution of the United States. The date each state adopted the Constitution tends to be taken as the date the Union itself was founded.
The city of New York was the federal capital for a year before the government moved to Philadelphia. In 1791, the states ratified the Bill of Rights, ten amendments to the federal Constitution that in theory prohibit the restriction of personal liberties and guarantee a series of legal protections. The Northern states abolished slavery between 1780 and 1804, leaving the slaveholders in the Southern states as defenders of the ” peculiar institution.” In 1800, the federal government moved to the newly founded Washington DC
In its quest to expand its territory westward, the state began a cycle of Indian wars that lasted until the late 19th century, stripping Native Americans of their lands. The purchase of Louisiana to France nearly doubled the size of the nation.  The 1812 war against Great Britain, which ended in a draw, helped strengthen American nationalism. The concept of Manifest Destiny became popular during this time.  The Treaty of Oregon, signed in 1846 with Great Britain, led the United States to take control of present-day Northwest America. The American intervention in Mexico in 1848 resulted in the loss by that country of California and much of present-day Southwest America. Fever of Gold of California from 1848 – 1849 further boosted the western migration. In half a century, up to 40 million buffalo were slaughtered for hides and meat and to facilitate the spread of railways. The loss of these animals, a fundamental economic resource for the Plains Indians, was an existential blow to native cultures.