Does anyone know the susceptibility of STM32F4 VBATT pin to ESD surges when changing out the RTC coin cell battery, or if there is some sort of internal protection against this happening such that an ESD protection diode would not be necessary?

[LMI2]

Data sheet of the CPU probably tells that. Google

[TDK]

All pins are generally well protected (in my opinion) against ESD, nothing is particularly sensitive to damage. However, you can still damage them if the shock is big enough. You may need to add additional protection if necessary.

ESD on VBAT is up to 1000V human body model.

[JRigh.1]

Much appreciated TDK! I missed that note trying to find something about it in the data sheet.