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There are two parts to buying a diamond ring. The diamond and the ring mount.
As for the diamond, you should definitely buy one online. But make sure you buy from a reputed company and get a diamond which has been certified by a top notch lab. Go for GIA preferably. They are the best. You can read the certificate no on the girdle using a basic loop and this will ensure you don't get duped. Buying online ensures massive range and better prices. Unless you are buying a low quality stone like SI2 or below there is nothing to see and inspect. But if you are then you should get it inspected for eye clean.
Stores cannot match the prices online unless they are conning you with a sub standard certificate. And it's unlikely that they will have enough inventory to show you a lot of actual pieces. They will have a few but buying from the lot is like buying directly from an airline and not comparing across airlines.
As for the mount, you might want to try it out to see how it looks on your finger in case you are buying an expensive studded one. If it's a plain one then go ahead and buy online. This is where the store has an advantage.
To give you the best of both worlds, the likes of Caratlane and bluestone have both online and offline buying options in the form of physical stores. Walk into one of the stores. Try out the mounts and then order the diamond online