In general, the United Nations, by means of the International Telecommunications Union, coordinates orbital assignments between member nations [1]. In the US, the FCC mediates for the American satellites to the ITU, so the FCC, as a member of the ITU, approves satellite orbital assignments [2]. In addition, the FAA plays a role in approving satellite launches as a matter of air traffic and safety, etc. [3]. The FAA doesn't get involved once the satellite is in space though.

As an example, one US company got in legal hot water by trying to skirt around the FCC in getting its satellites to orbit [4].

[1]. https://www.itu.int/en/plenipotentiary/2014/newsroom/Documen...

[2]. https://www.fcc.gov/general/international-bureau-satellite-d...

[3]. https://www.faa.gov/space/legislation_regulation_guidance/

[4]. https://qz.com/1503575/swarm-technologies-settles-illegal-la...