DS3231 Real Time Clock i2c 3volt battery 5V arduino
This is my understanding of the question:


  I know I can power DS3231 with +5V.  But If I disconnect the Vcc to work only with 3V battery (CR2032), will I be able to communicate with the RTC?


If the Vcc is powered down and the RTC is powered only from the backup battery, the microcontroller (MCU) will not be able to communicate with it through I2C.  I'm guessing that the MCU and RTC are powered from the same rail.  So, when Vcc is powered down, the MCU itself is powered down, and it wouldn't be communicating.