The board will never not power itself. If the voltage RAW is too low, the regulator output will be too low.
If the voltage VCC (the output of the regulator, of if applied directly) is below the Brownout Detector level (set by fuses), then a brown out reset will be asserted and the '328P chip won't start up.
A low supply voltage will not damage the Arduino. It will either not operate (brown out) or it may operate erratically (voltage too low per the Speed chart for the clock speed being operated at). It the internal clock is used at 1 MHz, or the clock divider fuses are set to operate at 1 MHz, then the voltage can be pretty low.
If you want power not to be applied if the supply voltage gets too low, then a low voltage detector controlling the Gate of a P-channel MOSFET is the way to go. That may be perhaps a N-channel MOSFET whose gate is driven by the supply voltage, when high enough it will turn on and bring the P-channel MOSFET gate low to turn it on.