Setting a Pin to HIGH to simulate a button with a "pull up" resistor in the below Fritzing diagram, I have a button that has a pull-up resistor, which when pressed, sets the GPIO Pin 23 to HIGH. On the Raspberry Pi, when the button is released, the GPIO goes LOW. The Raspberry Pi detects when the button is released and pressed. This is my desired behavior. 

However, in my current configuration, I have PIN 13 connected to the Base of a NPN transistor to try to simulate the same behavior as the button with a pull-up resistor. When pin 13 is set to high, it sets the GPIO 22 Pin to high. When I set the Arduino Pin 13 to low, The GPIO Pin goes low but the Raspberry PI does not detect that GPIO 22 pin went low. Is there a way to simulate the same behavior as a button with a pull-up resistor with an OUTPUT pin from the arduino?
First of all, I don't think you will need full simulation of a button. There are a couple of things this approach won't have:


Bouncing: it'll be (kinda, minus interference) a square wave. With a real button, the metal contacts "bounce," thus creating a weird waveform. Depending on interference, you'll still need debouncing code, although I would be surprised if it was necessary. I'd inlcude it in case I modified it to add a real button and/or there's some weird interference.
The time it's "pushed" may be very short, depending on your code. Keep this in mind when designing this.


Basically, you're hooking up the Arduino and the RPi together, with a voltage divider or a level shifter between the two.

Voltage divider:


  


Sparkfun

That is a voltage regulator (drawn four different ways; it's the same circuit). I calculated the values you will need: 1700 ohm for R1, 3300 ohm for R2.

Of course, serial (with USB of the Arduino) communication would be a lot easier...