It is not illegal to drive with a broken arm, but it is not particularly safe to drive with a sling or cast on the arm according to the American Academy of Orthopaedic Surgeons. The driver has to control the vehicle with one hand, and that puts both the driver and others on the road at greater risk.
A broken arm or wrist seriously impedes a person's ability to drive. The right hand controls shifting gears, which makes driving a vehicle with a manual transmission impossible with a right-arm injury. Most drivers primarily control the steering wheel with either their right or left hand, but the other hand is still critical in an emergency situation when it takes both hands on the wheel to control the car. Any restriction in one of the hands or arms reduces a driver's ability to steer and operate the car safely, and this affects the ability to conduct an emergency stop.