No. The US is a founding signatory to the Treaty on Principles Governing the Activities of States in the Exploration and Use of Outer Space, including the Moon and Other Celestial Bodies. That entered into effect a bit under two years before anyone set foot on the moon. Article two of the treaty states: "Outer space, including the moon and other celestial bodies, is not subject to national appropriation by claim of sovereignty, by means of use or occupation, or by any other means." So the US explicitly gave up any theoretical right to the moon before they got there to claim it.