Arduino + 6V Solenoid Valve
You don't power the solenoid by connecting it somehow to the arduino. You connect the circuit consisting of the solenoid, voltage limiting resistor, flyback diode and MOSFET in parallel with the arduino. You'd connect the +9 V to both the arduino DC positive and the solenoid, and you'd connect the ground to both the arduino DC negative and the MOSFET drain pin.
A solenoid which dissipates 1.6 W at 6 V will indeed draw 1.6/6 = 267 mA. However, using a 23 ohm resistor for dropping the 9 V supply voltage to 6 V won't work; 23*0.267 = ~6 V. You calculated the resistance of the solenoid, not the extra resistance needed for 9 V operation (which is 11.25 ohms, 0.8 W).  
That should be all you need, but I'd add a capacitor (0.1 - 10 uF) between the 9 V supply and ground for good measure, to prevent voltage transients from coupling to the microcontroller on the arduino board. Be sure to use a "logic level" MOSFET with a gate threshold voltage (Vgsth) below 3.5 V.  


In summary, the circuit I propose:  







simulate this circuit &ndash; Schematic created using CircuitLab