The specific question about why "the US" and not "the India" became popular from the Indian movie English-Vinglish. You can find the reference in this theatrical trailer.
To answer this question, we have to revisit our basics in articles' usage.
Articles are adjectives, usually used for common nouns. We do not use articles with Proper nouns. Example, an apple, an umbrella, a chair, a laptop, the clock, etc. We have singular article "a/an" used to describe a singularity of the common noun.
"the" is used to refer to a specific common noun. It could be used with singular and plural nouns. For example, the fastest route, the books in a library etc.
Coming back to our question, India is a proper noun. So we should not use an article with it.
America by itself is a proper noun. We should not use an article with it.
Let us consider the word "the United States of America." In this phrase, "states" is a noun, common noun.
If we consider the phrase "United states" only, "united" is an adjective describing the states.We are referring to a collection of states when using "united states". The word "united states" by itself is a common noun, because, there could be many united states, formed by a union of states.
In the American subcontinent, there is only one "united states". Hence, when we are referring to that particular united states in America, we should say "the united states of America".
Coincidentally, there is no other country in the world named as "united states". Hence, "the United States" now refers to the USA only.
