The title is somewhat misleading, in that it stands for what is more or less not to the document:
"The United States rejects any claims to sovereignty by any nation over outer space or celestial bodies, or any portion thereof, and rejects any limitations on the fundamental right of the United States to operate in an acquire data from space."
-Fas.org, U.S. National Space Policy
There doesn't appear to be any section in the policy to suggest that the U.S. would disallow themselves from taking control of aspects of space, as the assumption would be that the U.S. wouldn't and thus shouldn't mention it, or they would deem themselves empowered to seize aspects of space to fight... whichever war that may be plaguing the world at any time. Of course... how will space (likely the celestial bodies) be shared among humanity in research? With spatial commercialization, as well as war, establishing how research entities far beyond Earth's atmosphere function and who would be in charge of it. I theorize the UN or a similar power with the idea of representing Earth as a whole would be in charge of it, but a solution that simple would be miraculous in the event it held out in the long run. If we figure out how to efficiently colonize, or simply establish a colony, regardless of the initial cost, that has the capability of eventually producing its worth, then there's doubtless going to be debate about the power structure out there. It might not be a bad idea for such establishments to become self-governing, if not self-sufficient for awhile, in the event of a permanent presence.
In the end, I believe that there will be ownership issues to sort out that will likely come to be in the long-term, if it comes to be at all. Establishment of elsewhere in space will have politics tied in, but in what way may or may not be a total mystery from the thoughts at the present. One of my favorite authors, John Scalzi, had very interesting scenarios in his trilogy of Old Man's War, The Ghost Brigades, and especially the more politically focussed and final The Last Colony. In these books, humanity and its numerous colonies are all tightly governed by a greater power, the Colonial Union. The CU is technically all-powerful within humanity, as it is truly the power controlling the links to all colonies. They control communications, as well as having the definitive Department of Colonization (DoC) and the Colonial Defense Forces (CDF). Alltogether, the CU has total control of humanity in space matters (which is most of what concerns humanity, what with there being hundreds of other major sentient, space-faring races, many of which are actively at war with humanity). The CU, being all-controlling, has very important ups and downs. Some of the negatives include vice-like grip on information flow and massive secrecy, extremely aggressive space diplomacy, constant lying about greater intentions in space, non-democratic system, etc. Positives, however, include greater coordination of humanity's power as a result of being the sole power (no internal wars occur on an interplanetary scale, technology is for the most part freely circulating, etc). These books include massively advanced technology, including the ability to use multiple universes to instantaneously travel from one place to another, actually transferring conciousness from one body to another, nanotechnology that can assemble into incredible constructs extemely quickly, and the like. So given advances humanity may never have, how would we manage the power?