The US has ratified the
Outer Space Treaty, so we don't own the Moon (the plaque on the lander says "We came in peace for all mankind").
Articles II and III of the treaty are pertinent:
Article II
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.
Article III
States Parties to the Treaty shall carry on activities in the exploration and use of outer space, including the Moon and other celestial bodies, in accordance with international law, including the Charter of the United Nations, in the interest of maintaining international peace and security and promoting international co-operation and understanding.
By Article II, the US cannot make a sovereign claim to an an asteroid and assign mineral rights as it does on other federal lands, unless it abrogates the treaty. By Article III, "use of ... celestial bodies, in accordance with international law", it seems similar to mining on the sea floor in international waters, which is governed by an international authority.