In an RC circuit, current flows from the battery until the capacitor is completely

NO, the energies are not equal.

Explanation:

The total energy supplied by the battery is NOT equal to the total energy stored by the capacitor.

The total energy stored in the capacitor (given by the expression ), is exactly half of the energy which the battery supplies (QV), independent of the resistance, R.

(where Q is the charge flowing through the capacitor and V is the voltage on the capacitor).

The storage of energy on the capacity is the work done in transporting charge from one capacitor plate to the other, against the electrical forces. The charge differential in work done is what gives the total energy stored in the capacitor.