While the RIOT-002 adopts the RP605xxx4x with a minimum operating voltage of 2.4 V, the minimum voltage of the secondary battery is 1.8 V. Does the RIOT-002 stop operating if the voltage falls below 2.4V?

RIoT Environment Sensing Board

No, the RP605 itself continues operating until the input voltage reaches the under-voltage lockout detection voltage (Typ. 1.5 V).
However, as the accuracy of the battery monitoring function cannot be warranted, the recommended voltage range is 2.4 V or more.
The board itself operates until 1.9 V as the power good function of the R1801 outputs a low signal at the output voltage of Typ. 1.9 V or less.