Answer:
There are two meanings to fly, so for clarity's sake.
1) If you mean fly in the transitive (i.e. pilot an aircraft), the answer is unequivocally "no". You need at minimum a pilot's license.
2) If you mean fly in the intransitive sense (i.e. sit in a chair in a plane as a passenger), there is minimal screening in US domestic flights. Most Americans fly around the country using only their driver's licenses as identification, and it would follow that a British license would perform the same function. However, it is always advisable to use a Passport if possible, because there is no chance that the inspectors would turn that down.