As a general rule of thumb, one would normally treat any acronym as the words it substitutes for. However, there are many exceptions to this, which I will go on to detail.
For your specific example though, you would correctly say:
The United States of America
So it would be correct to say:
There are lots of other countries whose names contain the definite article, for example, The Bahamas, The Netherlands, The United Kingdom. When "United Kingdom" is abbreviated to "UK" we normally say "the UK". "The" is part of the name of The United States of America.
However, there are plenty of situations where the definite article is not required. For example, in international sports matches, it is common to say "UK versus USA".
Also, not all acronyms are treated this way. For example NASA is technically "The National Aeronautics and Space Administration", yet the abbreviation is never used with the definite article:
A hydrogen leak forced NASA to ground the space shuttle.
Other exceptions that spring to mind are acronyms which require a modified indefinite article than the words they substitute for - for example I would say "a light-emitting diode", but "an LED". And of course, there are acronyms which are have become accepted terms and are no longer written as acronyms, such as "laser".
So there is no hard and fast rule, but in text where grammar is otherwise important I would always include the definite article. In other contexts, it would be a case of consistency and whatever sounds right.