When an isosceles right triangle with legs measuring 4 inches is rotated about one of its legs, it forms a cone. The height of the cone corresponds to the leg around which it is rotated, which is 4 inches, and the length of the other leg becomes the radius of the base of the cone.
Since the base of the triangle has a length of 4 inches, when rotated, this leg serves as the radius of the base of the cone, which results in a diameter of:
\[ \text{Diameter} = 2 \times \text{Radius} = 2 \times 4 = 8 \text{ inches} \]
Therefore, the object formed is a cone with a diameter of 8 inches.