The problem here, is that the word "Organic" has two different meanings; one which is commonly understood in the marketplace, and another (probably older) meaning in the chemistry world. They are NOT the same.
The marketplace meaning of the term "Organic" refers a product that is produced totally from living organisms. Examples include wholegrain bread without artificial preservatives, beef raised without antibiotics, cotton clothing raised without pesticides, etc. Absolutely no nail enhancements fall into this category.
The chemistry meaning of the term "Organic" refers to all complex compounds of carbon, as Geeg correctly pointed out. Why? In the early 1800s, all known carbon compounds were derived from living organisms -- plants, animals, or their remains (manure, coal, petroleum, etc.) This is why this field of study was originally called organic chemistry. Since that time, many new carbon compounds have been created that were never found in nature, but the name organic chemistry has persisted, partly because scientifically it makes sense to study all the complex carbon compounds together in one category, and partly because the term is just too old and too entrenched to be changed. (A newer term, "biochemistry", was later created to describe the chemical reactions of life only. But nobody wants to go to the health food store and ask for a loaf of "biochemical" bread!)
Since acrylic and gel nails are made of complex carbon compounds, they fall under the research chemist's definition of the word organic. However, they are certainly not organic in the usual marketplace meaning of the word.
I remember one of their USP's for the product was that it would be perfectly safe if the client ate the nails!! I mean, I ask you!! 
