Will voltage or current determine a diode drop for this battery charging circuit?
The voltage across and current through the diode are not independent but are related by the diode IV curve.

And, since the diode is in series with the charger, the diode current will be limited to the maximum current the charger can supply.

So, for example, if the charger can provide a maximum of \$100\mathrm{mA}\$ of charging current, then the maximum voltage across the diode will be the given by the point on the IV curve for \$I = 100\mathrm{mA}\$.