GERD is Reflux disease, over production of acid in the stomach that can produce burns in the esophagus that can lead to Barrett's cellular changes to protect the tissues, and potentially esophageal cancers. Nexium is a proton pump inhibitor drug which also has an OTC version (its own previous incarnation) called Prilosec OTC, which reduces the amount of acid the stomach produces bringing things under control long enough for the burned tissues to heal, hopefully preventing the formation of Barrett's tissues in the esophagus. This particular cancer is one of the fastest growing in incidence the US today.
The acids produced by GERD can actually get high enough to affect the arynx and cause hoarseness. But overall I don't care for how this doctor has handled you. He shows no inclination to be concerned with tobacco cessation which the drug he didn't want to provide you with has a good track record of helping people quit. He was going to provide antibiotics without knowing what the infection might have been, it could have been different types of infection, requiring different types of antibiotics (gram negative and gram positive) but in the end he didn't even give you a broad spectrum one to cover things. (I would be against this in general because of the fact that it adds to your potential to develop bacteria that are resistant to antibiotics and you don't need that.) I think a new doctor and new opinions are in order.