"The moon belongs to America, and eagerly awaits the arrival of our astro-men."
-- The Simpsons

"Let anyone who despises the position of the moon get up and correct it."
-- Unidentified

Finders keepers.

The United States was the first nation to land on the moon. 1969. Neil Armstrong. Everybody knows that one.

Does that make it American property?

I don't believe that the world has a protocol for who owns undiscovered land anymore -- there isn't really any frontier left, and we are now sensitive to the needs of indigenous peoples. I suppose that if government ships happened upon an uninhabited, unclaimed tropical paradise, they'd probably hold onto it. But there's a fine line between a cay and a continent.

Reasonably, America can't claim the entire moon. Maybe the part that the Lunar Lander left marks on. Maybe where the flag stands.

This sounds like a debate. It's time to quote sources.

From NASA's "Ask the Space Scientist":

"No one does. It is by international treaty understood to be a free property. I do not believe that corporations or individuals may legally be allowed to own any part of it according to Space Law."
I swear, he actually said "Space Law." Original capitalization preserved.

While this is a very pretty answer (and reminiscent of the Kellog-Briand Pact), when a corporation (government, individual, whatever) acquires the resources to establish any kind of settlement on the moon, they will find a way to do it, and they sure as hell are going to claim the land that they build it on.

So all the high-falutin' principles of Space Law are essentially null and void in this case. While the particular U.N. statement that the Space Scientist refers to is just about universally ratified, who is really going to stop, say, Gillette, from building a moon colony? Nothing, if they can get their shit together.

The complete text of the treaty in question, "Treaty on Principles Governing the Activites of States in the Exploration and Use of Outer Space, including the Moon and Other Celestial Bodies", is available at http://www.oosa.unvienna.org/SpaceLaw/outersptxt.htm . Dated 1967.

Global Administration, or The Me-Me-Me Problem

In theory, the moon should be autonomous. A rotating group of nations could administer it and manage its everyday affairs. (Isn't idealism great?) There are a number of problems with this plan.

Nobody can ever agree on anything. Who, as a simple example, enforces the law on the moon? England? Russia? The United States? Zaire? What two nations have ever really been able to agree on a set of laws? This is a low-level problem, but it's one that goes all the way up. How about taxation? How about mutual agreements to bring people back and forth?

The US wouldn't allow it. The US, notably, does not recognize Antarctic claims made by any other nations. Antarctica is here, it's accessible, now. The official position on this is something to the effect of "You can use it, but we can throw you off if we feel like it." Any upstart nation that thinks that the United States government will sit back while they claim half of Luna has another think coming.

Property claims. If the moon becomes a habitable place, who owns the land? Entertainingly, "the Lunar Embassy" has been selling chunks of land on the moon for years. Thousands of people own one. They cost $16 and are 1,777 acres in size. This humorous aside illustrates the point. The world is past colonialism and cannot easily hand out prime real estate to wealthy landowners, or those privy to the secrets of the United Nations court. Disputes over land rights would last forever.

A rather lousy summary

The bottom line is that this is not an issue that will be arising anytime soon. We are fifteen years past the date when politicians predicted that we would have at least three settlements on the moon, and we haven't made much progress. The Columbia disaster will, of course, set things back even further. By the time that NASA or any foreign space agency has the resources to set up shop on the moon, the debate will probably have been resolved in an equitable fashion that only a skilled politician could dream up.

Until then, keep watching the skies.

Sources: "Ask the Space Scientist Archive." http://image.gsfc.nasa.gov/poetry/ask/a10913.html, available online at time of writing.
http://www.spacefuture.com is spectacularly useful for research on the subject.