I often get asked for "Organic Wine" (and it must say "Organic" on the label). Because the organic certification refers to the growing of the grapes not the production of the wine it cannot legally be labelled "Organic Wine" in Europe. It can be labelled "Wine made from organically grown grapes". (However in the USA this label means that only 70% of the grapes must be organic.)
Most smaller grower/producers use sustainable methods and avoid sprays. Their motivation is to produce the best, most natural wines and to sustain their vineyards for future generations.
There is little motivation to have "Organic" on the label as they market their wines on quality and reputation. If they happen to have gone 100% organic they won't be certified for 3 years and If they happen to have been certified it won't be splashed all over the front of the bottle. 
French wines may have a tiny logo half the size of a stamp on the back label, but that assumes they have a back label.
Wines marketed as organic tend to sell slightly faster (+1%) so there is a trend in supermarkets to get the word on the label where it is more important than the actual wine.
I recently read an article on ethical wine making by Master of Wine Susan McCraith* which illustrates some of the issues. One of the growers mentioned was Emmanuel Fourny, who makes the excellent Champagne we stock.
A few years ago he experimented with biodynamics on a couple of small parcels of land. Unfortunately mildew became so great a problem that they had to spray frequently with Copper Sulphate but still lost half of the crop.
Now here’s the thing. Copper Sulphate is seen as “elemental” and is allowed in organic and biodynamic viticulture. The problem is that Copper Sulphate does not biodegrade and can build up in the soil, stunting the vine and reducing the earthworm population. There are low residue, non-toxic, synthetic products available. Using these, although better for the environment, would mean no organic certification. So “Organic” is not always best and “Synthetic” is not always evil. As he wants to sustain healthy, biodiverse vineyards it is unlikely that Emmanuel will be marketing "Organic Champagne".
However, in the production of Veuve Fourny Champagne, Emmanuel uses organic fertiliser, lets the grass grow between the vines and uses pheromones in capsules (left) instead of insecticides to prevent female moths laying eggs on the grapes. He is also increasing the inherent health of the vine and using training methods to space out and aerate the bunches better to prevent mildew spreading. Most of the better small estates share this attitude.
Many growers in France are even reverting to using horse drawn ploughs to weed between row. Tractors are cheeper but they compact the soil which restricts weed growth to one or two species such as Colombine. Fewer weeds are dug into the soil so the vines begin to get hungry and may require chemical fertilisers then you are on the slippery slope. Clearing weeds entirely with herbicides can lead to the erotion of the fertile top soil as well as introducing unwelcome chemicals to the vine. Using chemicals in the vineyard innevitably leads to chemicals in the winemaking to counteract them. Pristine vineyards devoid of weeds may look attractive but the scruffy ones are healthier. (That is also the excuse I use for my overgrown flower beds)
So the message is this, just because it doesn’t say “Organic” on the label doesn’t mean they are not using eco-friendly practices. How can you tell? Well you can’t unless you ask someone who knows, like your friendly, neighbourhood, independent wine merchant.
A informative first hand account of biodynamic vine growing can be found in Monty Waldin's book, "Chateau Monty". (ISBN 978-1-906032-28-9)
Sustainable wine production: Growers also look to their wine making operations to be sustainable. This involves various aspects such as: reducing energy used in the wine making process, reducing the amount of water used to clean equipment, treating waste water and reusing it, using lighter weight bottles and recycled cardboard cartons.