No, gin will not break down into sugars or acids. Neat spirits, usually 40% or 37.5% ethyl alcohol can be regarded as bactericidal (causing death of bacteria) almost immediately upon contact. The vast majority of bacteria succumb within seconds.
In support of this, you need only think of Listerine, probably one of the most marketed and successful mouthwashes in the western hemisphere. Listerine contains around 27% ethyl alcohol the last time I looked and it's effectiveness in killing bacteria is down to this high concentration of alcohol. Gin as I said above, contains even more.
All the same, you should really get along to a dentist if the tooth is nagging you that much.