In rigid body rotation the applied force acts (you're pushing on the tire) a certain distance away from the axis of rotation (radius). The cross product of the applied force and radius give torque.
In this situation you are giving the wheel a torque. There is a centripetal force but it's effects are not immediately noticeable. The object is structured, and we use torque to describe its rotation. A centripetal force would occur if you tied a ball to a string and spun it in circles. The only obvious forces in rigid body rotation are the tangential force and radial force.
The radial force acts outward. The tangential force acts at a $90^o$ angle to the line of action. The line of action is the imaginary line through which your force acts. It is always connected to the axis of rotation. The vector sum of these forces is your net force.
So in summary there is a net inward force, it's just not as applicable in rigid-body rotation. Your net inward force (centripetal force) simply acts in the direction of change in velocity. This $\Delta v$ is inward along the circular path.