A country retains an identity when its name is changed. There are institutions in a country that exist independently of the name of the country. What processes would these institutions accept as proper to change the name of the United States of America?
Since the name "United States of America" is found (without formal establishment or definition) in the Declaration of Independence, and it pervades the Constitution, a statute establishing a different name for the country would probably be insufficient. A constitutional amendment should suffice, however.