No because the maximum voltage you can apply to an arduino pin is 0.5v ABOVE the supply voltage which is 5V because the regulator cuts the 12V down to 5v before it reaches the arduino.
More specifically...even though the cathode side of the LED+resistor arrangement is close to 0V when the rated current is flowing, it floats to 12V when no current is flowing. Your sinking switch still needs to be able to handle full voltage (and the Arduino pins cannot). If you never turned the switch off, you might be able to power the circuit that way, but wouldn't it be boring?