I contend Free Speech does not apply to a broadcast license. Not everyone can qualify for a broadcast license. It is identical to driving: not everyone can have a license to drive. To continue to drive, you agree to follow certain rules. It's not so much about free speech as responsibility involved in holding a license.
I doubt this will go anywhere, more of the FCC going through the usual steps when a complaint is filed and calling attention to it when they feel a number of complaints have been received.
While the FCC says obscenity is deemed such, in part, by community standards with the realization that those standards vary a little by community, how can the FCC determine what standards are in, say, Sioux City, El Paso, Los Angeles, New York and Birmingham, for example? What could pass in LA would make heads roll in another market. How might the FCC determine the specific community's standards? If that is true, a station could be fined in Birmingham but not in Los Angeles for the same 'bit' because community standards are a bit different.
I have always contended you can get away with much more in a major city because of the number of signal choices. In the small market you have a much wider cross-section of the population tuned to any station meaning the greater the chance of offending someone. If ratings for The Late Show are up, even following the 'incident' it would seem the 'masses' have spoken: meaning no foul here. I wonder if the complaint had any merit in the first place. In my opinion, it doesn't qualify as obscene, maybe not tasteful but certainly a ploy for better ratings.