Even if she worked for a private company, she can't be fired for making a complaint of what she perceives as illegal harassment. She has a Federally protected right to do so.
Let me put it this way. You CAN be fired for this because there is no law that says you can't be. However, just because you can doesn't mean you will. It sounds as if at worst, you may be asked to sign something agreeing to "go and sin no more". It is unlikely that it will require that you acknowledge harassing her.
For the love of Pete, though, if your co-worker is that sensitive, next time ask her if she wants the cough drop first. Or better yet, let her buy her own.

