You don't say in your question whether you're looking to cycle in France because you're a serious sporting cyclist, or just looking for a place where you can abandon the car, and move around on a bike instead.
If the latter is what you were looking for, I've got a couple of suggestions, though both will involve crossing small expanses of water first!
1. Ile de Porquerolles
This gorgeous island, beloved of the writer Georges Simenon, is a 20-minute boat trip from Hyeres, on France's Mediterranean coast.
Porquerolles is about 4 miles long, and 2 miles wide, and the island is covered in fragrant eucalyptus, juniper, fig and pine trees, as well as lavender, rosemary and myrtle.
Building on Porquerolles is very, very tightly-controlled - because the island has national park status. There is no motorised transport at all - apart from a mini-Moke to take luggage to the hotel - so cycling and walking are the only methods of travel. You can rent bikes on the island.
Here are some cyclists on Porquerolles, showing how it's done:
2. Ile de Re
The Ile de Re is situated off the western coast of France and is known as the "jewel of the Atlantic".
It is connected to the mainland at La Rochelle by the longest bridge in France:
The island has long, sandy beaches, oyster farms, and is perfect for fishing or cycling. Not cycling at speed, though, as there are cyclists everywhere!
Basically, the Ile de Re is a great mix of bohemian and genteel, the perfect place to chill out, and to cycle instead of driving.