The FAA does have licenses over launch and they are trying to impose rules for upper stages of launch vehicles [0]. The FAA said they would complete these regulations in 2025 [1], but I haven't seen something saying they have gone in to effect yet.
The FCC does deal with disposal requirements for US satellites that are launched. In order to secure a license from the FCC you have to prove that your satellite will meet the latest guideline that it will be disposed of (either de-orbit for LEO, or moved to disposal orbit for higher orbits) within 5 years after mission complete [2]. Unfortunately this doesn't seem to apply to upper stages for some reason even though I would say that it is an orbit object that gets licensed and would "complete the mission" after deploying the satellites and have to abide by the 5 year rule.
[0] https://www.faa.gov/newsroom/faa-proposed-rule-would-reduce-...
[1] https://spacenews.com/faa-to-complete-orbital-debris-upper-s...