On the one side, the USA is just one country. Logic says it should be, then, singular, just like the United Kingdom is. Example:
The USA owns this domain.
On the other side, if I however expand "the USA" to "the United States of America", I'd tend towards using plural — the noun the verb agrees with, "States," is definitely plural. Example:
The United States of America own this domain. → The USA own this domain.
What form should I prefer?
Short answer: in contemporary English, both USA and the long form United States of America are treated as singular nouns.
Long answer: Language Log has documented this in great detail. In the 18th and much of the 19th centuries United States was treated as plural, but in the latter half of the 19th century the singular usage became more common. Today, the singular usage is the only accepted usage, except for the case of a few fixed phrases. In fact, "in 1902 article in the Washington Post reported that Foster's work (which evidently was reprinted as a pamphlet) had persuaded the House of Representative's Committee on Revision of the Laws to rule that the United States should be treated as singular, not plural."
The United States of America own this domain
To me this sounds a little bit awkward, as the United States of America is one entity. Actually, it's likely because the pluralism is buried in the middle of the term.
If you were to use simply The United States... I would accept either own or owns, depending on what you're trying to emphasize: the collection of states as one entity or the collection as a group of states.
Both "USA" and "The United States of America" are a single proper noun. They are names. I don't believe you can point to a word within a name to call the name a plural. Both names refer to a single entity. They should be followed by the singular form.
The exception is in some British English where singular nouns representing collectives (companies, teams, governments, etc.) are treated as plural. Some reading on that: http://languagelog.ldc.upenn.edu/nll/?p=877
Very late additional answer:
As others have noted, today "the United States of America" is just about always treated as singular.
However, you will occasionally hear someone refer to "these United States of America", or say "the United States are ...". Sometimes this is done for political reasons: a hot subject of debate in the US is what the relative power of the federal government versus the state governments should be, and some who believe in greater autonomy for state governments will deliberately say "these United States" to emphasize the idea that there are, or should be, multiple sovereign state governments. More often it's because people want to sound "old timey".
Readers Digest magazine used to have -- I'm not sure if they still do, I haven't seen a copy in a while -- but they used to have a feature with jokes about American life that was called, "Life in These United States".
I think you chose whichever sounds best to you, as there's clearly not right or wrong. Style guides, for example those internally used by the BBC, suggest you pick one and stick with it, at least for that article/feature.
Agree with the form you're using; "the USA owns", "the United States of America own". The initialism doesn't need to be treated as if it were expanded.
As the great Civil War historian Shelby Foote put it, the War "... made us into an 'is'". Before the War the States were referred to in the plural; afterwards, it was painfully clear that they form one entity: the UNITED States, hence, one polity. Of course, within the United States itself, one could refer to more than one State in the plural, as they retain some individuality (State constitutions, legislatures etc.) within the national - and singular - whole.
