Compost by definition is organic. It is the decayed remainder of flora and fauna and thus is processed organic material.
That said, additives both known and unknown can add inorganic material to the mix.
Some organic gardeners like to add certain mineral additives to their compost such as dolomitic limestone or greensand to lower pH and add calcium to the mix. It is doubtful if this is more effective than adding these materials directly to the soil especially in the case of pH control as compost piles tend to go neutral by the nature of their composting action. Still, it means that both materials can be added to the soil in one go.
On the other hand, one has to be careful of the materials added to the compost pile. Grass clippings, fall leaves and other supplies that may be picked up at the roadside as others throw them out, can have pesticide and herbicide residue. Either leave the offerings where they lie or be sure that the people disposing of them have not used any chemical applications over the past month or more.
It is easiest to ensure that compost is organic by making one's own. There is no need for other than organic material in the first place. A healthy mix of brown and green vegetable matter will have almost all if not everyone of the minerals necessary for good plant growth. Green or wet material will usually be damp, odored, and some variation green in color. Examples are grass clippings and kitchen waste. Brown or dry materials, on the other hand, will be dry, brown or gray in color, and largely odorless. Autumn leaves, spoiled hay, wood shavings, and dried weeds are examples.
When gathering materials, a volume mix of about two parts brown to one part green matter is about right for a good pile. If there is much newspaper or wood shavings in the brown material, a ratio of one to one is good. You want enough carbon from the dry material to feed the microorganisms in the pile and enough nitrogen for them to produce protein.
Wood shavings are not the threat to compost piles that some composters believe them to be. In fact, many wood turners and other wood workers seek spalted wood for its color. Spalted wood is simply wood that has dark lines and tan areas in it from the beginnings of rot. Composting is just a name for controlled rotting so spalted wood is already underway. Look for shavings of maple, birch, and poplar in particular as these woods are very prone to spalt.
Mix the materials well and make sure that they are sufficiently moist. Ideally they will feel like a wrung out dishcloth. If the pile is well built it will heat in a couple of days to where it is too hot to keep your hand in it. The more often it is turned the faster it composts, but it will rot down for you into a rich compost. When it is dark, crumbly and earthy smelling, your compost is ready to use.
You can know the compost is organic easiest by making your own. Like most things, honest materials and honest work give honest results.