Up until about 1800 the majority of the area of North and South America was claimed to be part of a kingdom with the title of "The East and West Indias". And thus the aboriginal inhabitants of that kingdom were called "Indians" since they lived in one or another of "The Indias". According to Spanish colonial law, the aboriginal inhabitants of that vast kingdom were legally described as "Indians".
The American Indians were called Indians because that was their legal racial name, meaning aboriginal inhabitants of the Indias, and had been for centuries. Changing their racial name would probably have required officially changing the title of the gigantic kingdom that claimed most of the Americas to something else.
And calling the aboriginal peoples "Indians" was picked up by Europeans in Europe and in the neighboring Portuguese, French, Dutch, and English colonies.
In etymology, the study of words, it makes almost as much sense to call North and South America "India", and their natives "Indians", as it does to call The Republic of India "India" and its natives "Indians". The United States of America is almost as much "India" as the Republic of India is.
In the old days India did not mean the area of the Republic of India, nor the area of the Indian Empire or British Raj, nor the area of the Mughal Empire. It was a very vague geographical term. So vague that probably about half the land area of the world, and almost half of the oceans, has been part of one "India" or another at various times.
Medieval Europe had a geographical term of the Three Indias which had very vague geographical limits since very few medieval Europeans had ever been to any part of them.
The concept of the Three Indias was in common circulation in pre-industrial Europe. Greater India was the southern part of South Asia, Lesser India was the northern part of South Asia, and Middle India was the region near the Middle East. The Portuguese form (Portuguese: India Maior) was used at least since the mid-15th century. The term, which seems to have been used with variable precision, sometimes meant only the Indian subcontinent; Europeans used a variety of terms related to South Asia to designate the South Asian peninsula, including High India, Greater India, Exterior India and India aquosa.
However, in some accounts of European nautical voyages, Greater India (or India Major) extended from the Malabar Coast (present-day Kerala) to India extra Gangem (lit. "India, beyond the Ganges," but usually the East Indies, i.e. present-day Malay Archipelago) and India Minor, from Malabar to Sind. Farther India was sometimes used to cover all of modern Southeast Asia. Until the fourteenth century, India could also mean areas along the Red Sea, including Somalia, South Arabia, and Ethiopia (e.g., Diodorus of Sicily of the first century BCE says that "the Nile rises in India" and Marco Polo of the fourteenth century says that "Lesser India ... contains ... Abash [Abyssinia]")
So the largest definition of the Three Indias or Greater India or India would include all of South Asia and everything from Ethiopia to the Philippines, or possibly to China and Japan.
So when Columbus arrived at the Bahamas, Cuba, and Hispaniola in 1492 he believed that he had arrived in the Indias, in islands somewhere south of Japan, and only a few hundred or a few thousand miles from wealthy civilized lands like India or China.
Thus the members of various tribes and ethnic groups in the new lands began to be called Indians, because they were believed to be natives of a part of "East India".
In 1504 King Ferdinand II of Aragon, etc. added Lord of the Indias of the Ocean Sea to his titles.
Nos Ferdinandus Dei gratia Rex Aragonum, Siciliæ, citra et ultra Farum, Jerusalem, Valentiæ, Majoricarum, Sardiniæ, Corsicæ,
Dominus Indiorum maris Oceani,
Dux Athenarum et Neopatriæ,
Comes Roxilionis et Ceritaniæ,
Marchio Oristani et Goccani,
administrator et gobernator regnorum Castellæ, Legionis, Granatæ etc. pro Serenissima Regina Johanna, filia nostra carissima
Ferdinand II renounced his rights in America to his daughter Queen Joan I of Castile, etc. by the Treaty of Vilafafila in 1506 and dropped the Indias from his title.
In 1506 Queen Joan I of Castile, etc. and her husband King Philip I added the title of King and Queen of the Indias, the Islands and Mainland of the Ocean Sea, to their titles.
Don Felipe e dona Joana, por la gracia de Dios, rey e reyna de Castilla, de Leon, de Granada, de Toledo, de Galizia, de Sevilla, de Cordoba, de Murçia, de Jahen, de los Algarbes, de Algezira, de Gibraltar e de las Yslas de Canaria y de las Yndias, Yslas y Tierra Firme del Mar Oçeano,
principes de Aragon e de las Dos Seçilias, de Jerusalem,
archiduques de Austria,
duques de Borgoña e de Brabante etc.,
condes de Flandes e de Tirol, etc.,
señores de Bizcaya e de Molina etc.
I am a little uncertain whether the Indias and the Islands and Mainland of the Ocean sea were supposed to be two separate kingdoms, or a single Kingdom of the Indias, the Islands, and the Mainland of the Ocean sea.
In 1581 "Indias" was replaced with "East and West Indias" so the title became "East and West Indias, the Islands and Mainland of the Ocean sea". The "East Indias" were the Philippines and other islands in East Asia, often called the "East Indies" in English, and some Pacific islands, and the "West Indias" were all the islands, sometimes called the "West Indies" in English, and mainlands in the Americas.
The term "West Indies" for Caribbean islands is the reason why a movie about fictional female pirate Anne Providence in the Caribbean is titled Anne of the Indies (1951).
Don Phelipe, por la gracia de Dios, Rey de Castilla, de Leon, de Aragon, de las Dos Sicilias, de Jerusalen, de Portugal, de Navarra, de Granada, de Toledo, de Valencia, de Galicia, de Mallorcas, de Sevilla, de Cerdeña, de Córdoua, de Córcega, de Murcia, de Jahen, de los Algarues, de Algecira, de Gibraltar, de las islas de Canaria, de las Indias orientales y occidentales, Islas y tierra firme del mar Océano,
archiduque de Austria,
duque de Borgoña y de Brauante y Milan,
conde de Abspurg, de Flandes y de Tirol y de Barcelona,
Señor de Vizcaya y de Molina, etc.
Joseph Bonaparte, the monarch from 1808-1813, used the short title of "King of the Spains, the Indias".
Don Josef Napoleón por la gracia de Dios y la Constitución del Estado Rey de las Españas y de las Indias
And in 1813 the old title was readopted, including "King of the East & West Indias, the Islands and Mainland in the Ocean sea".
Don Fernando, por la Gracia de Dios, Rey de Castilla, de Leon, de Aragon, de las Dos Sicilias, de Jerusalen, de Navarra, de Granada, de Toledo, de Valencia, de Galicia, de Mallorca, de Menorca, de Sevilla, de Cerdena, de Córdoba, de Córcega, de Murcia, de Jaen, de los Algarbes, de Algeciras, de Gibraltar, de las Islas de Canaria, de las Indias Orientales y Occidentales, Islas y Tierra firme del mar Océano;
Archiduque de Austria;
Duque de Borgona, de Brabante y de Milan;
Conde de Abspurg, Flándes, Tirol y Barcelona;
Señor de Vizcaya y de Molina, &c.
The Kingdom of the Indias ruled the largest area ever in any kingdom ruled by Europeans, and claimed a much larger area, all of the American continents except for eastern Brazil and eastern Greenland, and also claimed rule over most of the Pacific Ocean.
And since the official name of that vast region was the Indias, it was only natural that the native peoples as a whole would be called Indios, "Indians", and that English speakers copied that name.
In short, any person residing in the western hemisphere has almost as much right to be called an Indian as any person residing in the Republic of India.
The Heraldry of the Kingdom of the east and West Indias
Here is a link to a discussion of the coats of arms of the Spanish Kingdom of the Indias.
I may add another coat of arms for the Indias. The Austrian Candidate for the Spanish throne in the War of the Spanish Succession (1701-1714), Archduke Charles, lost, but became Emperor Charles VI in 1711. By treaty he retained the right to use his Spanish titles during his lifetime. So his full title was:
Emperor of the Romans;
King in Germany, of Castile, Leon, Aragon, both Sicilies, Jerusalem, Hungary, Bohemia, Dalmatia, Croatia, Slavonia, Navarra, Grenada, Toledo, Valencia, Galicia, Majorca, Sevilla, Sardinia, Cordova, Corsica, Murcia, Jaen, the Algarve, Algeciras, Gibraltar, the Canary Islands, East & West Indias, the Islands & Mainland of the Ocean sea;
Archduke of Austria;
Duke of Burgundy, Brabant, Milan, Styria, Carinthia, Carniola, Limburg, Luxemburg, Gelderland, Württemberg, the Upper & Lower Silesia, Calabria, Athens, Neopatria;
Prince of Swabia, Catalonia, Asturia;
Margrave of the Holy Roman Empire, of Burgau, Moravia, the Upper & Lower Lusatia;
Princely Count of Habsburg, Flanders, Tyrol, Ferrette, Kyburg, Gorizia, Artois;
Landgrave of Alsace;
Margrave of Oristano;
Count of Goceano, Namur, Roussillon, Cerdagne;
Lord of the Wendish March, Pordenone, Biscay, Molina, Salins, Tripoli, Mechelen;
Thus the successors of Emperor Charles VI sometimes used Spanish coats of arms among their many coats of arms. Including one which can be blazoned: "Azure (blue) a lion rampant (rearing up) argent (silver) with a cross or (gold) in the dexter (right) paw". This coat of arms is called "India" and no doubt refers to "the Kingdom of the East and West Indias, the Islands and Mainland of the Ocean Sea".