Surely the OP was enquiring about the lack of fixed gear bikes on the road. Whereas Carbon in itself is a fantastic material for road use, frames made out of carbon tend to have either vertical drop-outs or are track specific (heavy-ish, very stiff).
There is no reason why you cant ride a carbon track bike on the road, but spending a grand or so on a frame poorly designed for the purpose seems a bit silly.
Surely the OP was enquiring about the lack of fixed gear bikes on the road. Whereas Carbon in itself is a fantastic material for road use, frames made out of carbon tend to have either vertical drop-outs or are track specific (heavy-ish, very stiff).
There is no reason why you cant ride a carbon track bike on the road, but spending a grand or so on a frame poorly designed for the purpose seems a bit silly.
Note tube diameters of bikes below.