No, the FPGA device would not automatically start reconfiguration when CRC error check detects soft errors during user mode. The CRC error detection process will continue until the device is reset by pulling the nCONFIG pin to low. The nCONFIG pin must be pulsed low to high in order for the reconfiguration to start. This should be performed by an external controller.
Refer to the respective device handbook for the minimum time specification you should hold the nCONFIG pin low.