The diode voltage drop will be determined entirely and only by the current flowing in it.
The current flowing will be set by the overall circuit and may be affected by the diode voltage drop.
So the two will interact even though the actual diode voltage is solely diode current dependent.
However: A key question is WHY? do you have a series diode.
The result of including one depends on what charger IC you are using (which you should have told us) and the circuit (which you should have shown us) but usually this means that the battery will be very substantially low on deliverable capacity (and long on cycle life). It also MAY mean that the charger IC will not change from CC to CV mode ever and will keep trying to charge the battery indefinitely.
Unless there is a superbly good reason to do this and you or the person who designed the circuit know exactly what they are doing then it is a "bad" [tm] thing to do.
Worst case - and this depends on the circuit - the charger IC may not swap from CC to CV mode, as the battery voltage that it sees appears too low, the battery will approach 4.2 V with diminishing current but never reach it, so charging will continue as the battery slowly asymptotes to 4.2V (as current falls Vdiode drops) and if you leave the battery connected it will be damaged and its lifetime greatly shortened.
If the charger IC has a timeout timer it may prevent strange happenings.
What is the charger IC?
Show us the circuit diagram.
Why are you using a series diode?
A series diode will prevent (almost) back feeding of battery into the charger IC when the power to the charger IC is removed. Any IC worth its salt will be designed to handle this and will typically draw around 1 uA (maybe 5 uA worst case for good ICs). In most scenarios this is acceptable. 1 uA = 0.168 mAh/week. 10 uA = 1.68 mAh/week.
10 uA = 88 mAh/year, so if you want very long periods between charging a 10 uA drain may be untenable.
A series diode will protect against reverse battery connection but is not a good solution due to the problems discussed above.
If back feeding prevention and/or battery polarity protection is desired a MOSFET can be configured to do both jobs. Ask if relevant.
In summary, you are trying to achieve a relatively minor circuit feature which could be obtained in some other manner by playing fast and loose with the LiIon charging algorithms in a way that no circuit ever used commercially attempts to do. The results can be expected to be bad.
You need to say WHY you wish to use PGOOD in this manner and wish to avoid "doing it properly". Reasons may include lack of space, lowest cost or not enough knowledge of hos to do this or ... . Until people know why and what you want solutions are unlikely to be 100% good.
The added circuit diagram shows the battery supplied by the series output diode and the IC's battery voltage sense feedback divider connected prior to the battery. this means that the battery voltage will be a diode drop lower than what the IC "sees" - OR the IC will see a voltage a diode drop higher than the battery actually is. This is bad.
An alternative is to connect the feedback divider at the battery, outside the diode.
This is also bad.
The reasons are covered more or less in the notes I've written previously.
LiIon batteries are reasonably easy to charge if done correctly. But they are quite easy to damage if charged incorrectly. A crucial part of part of charging is to know when to swap from CC (constant current) to CV charging. This occurs at Vbattery = 4.2V - with possible adjustment for temperature. Adding a diode as shown will cause change to CV mode when the battery is at about 3.9V. This is well before completion of the CC phase and will lead to an early slow down of charging. As battery voltage slowly increases Icharge will drop and lower current leads to lower Vdiode so as Ichg drops to near zero I diode will drop to near zero and ultimately the battery will get to almost 4.2V.This will take MUCH longer and it's possible that the IC will not change to CV mode at all (as perceived Vbat is too low). This means the battery will be 'floated" at somewhere in the 4.0 - 4.15 range. It will have significantly lower capacity AND may be damaged.
Also, without wading carefully through the data sheet and/or playing with the IC itself I can't be sure, it MAY be that the IC not seeing a battery will cause it to start up in the trickle start "dead battery" mode.
I mentioned the "divider outside diode case in my prior notes. It has its own issues and, as the IC, sees the actual battery voltage it probably fails your PGOOD requirement.
The confusion as to whether
diode current sets diode voltage or
diode voltage sets diode current
is caused by the fact that neither does either in a real-world circuit.
Diode current and voltage are set by a known relationship.
Ignoring second order effects (eg temperature), then
If you know V then you know I.
If you know I then you know V.
IF you connect the diode across an ideal voltage source then V does control I.
If you connect the diode across an ideal current source then I does control V.
BUT if you connect the diode in a real-world circuit I and V vary as the circuit adapts to what it "sees" until a steady state is reached (or it oscillates in some cases).
The diode I & V conform to the devices I & V curve.