That circuit would cut off at a voltage set by the 10 kOhm potentiometer. It would have some hysteresis, meaning that it would only function as on or off, and never be partially on.
There is no current limit, but that might not be a problem, depending on what the Lithium Ion charger does.
You need a separate 12 V supply. There are other ICs that you could use that would allow the op-amp to run from the 5 V supply.
The range of adjustment for the potentiometer is huge. If R1 were changed to 30 kOhm, then the 10 kOhm potentiometer would give a range of from about 3 V up to about 4 V.
The amount of hysteresis provided by R3 depends on where the potentiometer is set.
What are you trying to do with the circuit? Are you trying to control the charge, or to stop charging at a pre-set voltage?
Charging Lithium-Ion cells does need to follow the rules, but they aren't very complicated.
1) Keep the voltage below the maximum for the cells. It is often 4.2 V. (and watch all the cell voltages if charging a battery with cells in series). Lower voltages give less capacity, but longer cell life.
2) Keep the current below about 1 C, so don't try to charge the batteries faster than about 1 hour. Slower is fine.
3) Stop charging when the voltage has hit the maximum and the current falls below about 0.05 C
4) Do not charge the cells outside of the temperature range 5 - 40 deg C
If you read the
L6924D datasheet it will show you a lot about how chargers function.