The first issue is liability as prescribed under the Outer Space Treaty of 1967. Normally, the launching country is liable for any damage done in the event of an accident. When the treaty was written, very few private companies were launching satellites. Instead, governments did that, so the governments are liable. This gives the government at least some say-so when it comes to launches.

Now, Article VII of the Outer Space Treaty says that the liability falls on the launching nation. However, since the Russian Space Agency has very little money, you can count on the lawyers going after America because we have deeper pockets, probably based on ownership as defined in Article VIII.

The other issue, and perhaps more to the point, has to do with laws regulating technology transfer between American companies and countries like Russia. Even though the Cold War has ended and American companies like Sea Launch and Lockheed-Martin are importing Zenit boosters and RD-180 rocket engines, there are still restrictions on sending technology the other direction. These restrictions might have to be addressed to bring them up to date, but until then, both sides have to comply with existing law.

The point about getting approval to launch to the moon might be more of getting approval by boards from the National Academy of Sciences or something like that rather than formal government approval. According to Article II of the Other Space Treaty and other agreements, no nation is allowed to make any territorial claims on "the moon or other celestial bodies", but there may be provisions to coordinate projects to prevent damage like contamination. I don't know the specifics of such restrictions, I'm just speculating as to their origin.