1. Once the capacitor charges, the op-amp is unable to provide more voltage, and it stops charging (it's not an ideal current source, of course). If you didn't have a switch and comparator connected, this will be the final state.
2. You don't want a delay -- instead, use a comparator with hysteresis, or a bistable circuit (e.g., examine the internal diagram of the 555 timer). That way, the capacitor is not discharged to an unknown voltage, but the discharge stops when the capacitor is discharged below the threshold.
Tim