Why wouldn't they air it?For example, if Kirk was going to kiss Uhura, then he should've really laid one on her for several seconds full screen, and Roddenberry should've left it to the network to explain publicly why that was something they wouldn't air.
On August 17th of 1965, NBC vice president Mort Werner sent Roddenberry a letter reminding Roddenberry of NBC's employment policy of including minorities, specially "negros," in NBC shows, and asking for Roddenberry's cooperation.
I've never read anything about NBC having a problem with "the kiss."
At least from what I understand from memory (and someone chime in if I'm wrong), when NBC found out the kiss was in the script, they were concerned with the reaction it may create in their southern affiliates. NBC eventually said two shots of the scene had to be done, one with the kiss and one without. According to Nichols, Shatner deliberately ruined all of the non-kiss takes so that the only usable shot was of the kiss. NBC finally told them to go ahead with it. Nichols speculated that part of the reason why they relented may have been they knew the series was going to be cancelled, anyway.
Edited to add: After a short search, I found a link to a Nichols interview about "the kiss." There may be other versions, but what's in here is in most versions.
http://www.geeksofdoom.com/2012/09/...talks-about-her-famous-kiss-with-captain-kirk