A driver with 20 Stamina won't drop health over the length of a race. If he starts out at 95%, he finishes with 95% health. If he has 19 Stamina, his health drops around 0.1% a lap.
Does depend on the push level though. If a drivers is on push 4/5 all race for example in Monaco, also the 20 stamina drivers will drop some health over the length of the race. But, it sure will be less than a lower stamina driver