Yes an acid can cause corrosion.
It depends on the material considered and the concentration of the acid.
The reaction involved in the corrosion process in acidic solution is the cathodic reduction of Hydrogen ion H+ to give Hydrogen gas H2.
H+(solution) + electron -----> H2 (gas)
the metal corrosion reaction is the anodic oxidation:
Metal -------> Metal n+ + n electrons
This is for example the corrosion process of Iron or Carbon Steel in contact with an acidic solution.
Stainless Steel or Nickel alloy are more resistant to acid solution and are corrored only by some acids.
Acids results in over $12 billion of corrosion related costs a year in mining, pulp & paper manufacturing, petroleum refining, and chemical manufacturing. Acid resistant plating (high phosphorus electroless nickel) is commonly used to limit corrosion in highly acidic environments.