A similar question might be why doesn't the air drag on a sky diver push them back up higher?
When the skydiver leaves the plane, air drag is minimal. Gravity accelerates them downward. As they fall faster, drag increases until it has a magnitude equal to gravity and acceleration stops. The diver is at terminal velocity. The force of the drag stops the acceleration, not the fall.
When circuit is closed, the voltage difference from the battery causes the current in the wire to increase. As the current rises, the potential dropped across the resistor increases as well. When the potential equals that of the battery, the current has reached the maximum and the circuit is at steady-state. The resistance in the battery prevents the current from increasing further, but does not push it backward.