Who else besides the FCC needs to approve SpaceX launching over 4,000 satellites into space? That seems like a fairly significant number of satellites, but admittedly I am a layman in this area.
The ITU as well. Part of their proposal involves reuse of spectrum currently allocated to geostationary satellites, and the mechanism for noninterference is physical separation: while they may transmit on the same color as a GEO-sat, they won't be simultaneously visible from a normal antenna for GEO use as they will point in a different direction. The GEO-sats (and their spectrum allocation) is confined to the ecliptic, while these LEO-sats will be transmitting from any direction except overlapping the ecliptic.
It is a pretty significant number of satellites. For whatever reason the FCC has taken up the role of regulating satellite collisions and such, and if you go dig up SpaceX's fillings you can see them discussing mitigating issues related to this (keeping probability of a collision for the entire constellation less than the acceptable number for a single satellite, making sure the satellites burn up on re-entry to mitigate "a satellite fell on my head" risks, etc).
Each country more or less is responsible for regulating itself. The International Telegraph Union sometimes comes into play for spectrum issues, but I think most of their influence is in GEO since they handle slotting there.
I'm curious specifically about the prospect of circumventing firewalls in China or other places where internet access is heavily restricted and censored. I imagine that China could tell the US government to tell SpaceX to cut it out and cite various relevant international treaties, but in a lot of ways it's more of a political question than a legal one. I doubt many people outside the North Korean leadership would object to providing Internet access to North Koreans, for instance.
China has the ability to shoot satellites out of the sky. Now if they have the political will to do so remains to be seen. But they can also easily jam them in their own backyard.
