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].
The trick is that they don't just need the FCC for the orbit clearances the FCC also deals with spectrum management in the US and Starlink needs similar approval to have (legal) nodes in any country they want to have customers in.
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...