since the ground clearance of a sports car is low. Do we use small tires for them?
Ground clearance has nothing to do with either wheel diameter or tyre profile. It is determined by suspension geometry and the clearance allowed by the body (particularly wheel arches).
Often low profile tyres are more about cosmetic appearance than actual performance. Note that F1 cars have tyres with a larger aspect ratio than many standard road cars and still achieve small (and in some conditions zero) ground clearance.
There is also an important difference between ride height and effective ground clearance. Ride height is simply the distance between the floor of the car and the road ground clearance determines the ability to traverse ramps and bumps without grounding and is affected by suspension travel and wheelbase. Short wheelbase vehicles will be less likely to ground on very rough terrain than longer wheelbase ones with equivalent ride height.
The placement and vulnerability of components such as prop shafts axles and sumps will also have an effect on effective ground clearance. In fact some extreme off road vehicles have steel or composite plates protecting their undersides to allow them to slide over rocks without breaking anything important and similarly may have independent wishbone suspension at both front and rear compared to standard models with a solid rear axle.
Low profile tyres will tend to make the sidewall of the tyre stiffer which can be beneficial for traction and braking although this must be considered in the context of the overall suspension design as sidewall flex has an effect on both ride comfort and steering feel.
Low profile tyres can also allow for larger diameter brakes and more space for brake cooling ducts although this can also increase the unsprung mass of the wheels.