Technically speaking, white on an LCD monitor or screen is the color displayed when the Red, Green, and Blue value of a given pixel are all the same and at or near saturation. With a 24 bit color space like the vast majority of computer monitors use, this would mean a value of 255, 255, 255 (8 bits per color; 2^8 = 256 different intensities = 0-255 inclusive). A value of 128, 128, 128 would render neutral gray. A value of 0, 0, 0 would result in black. If a monitor is calibrated and profiled properly, it will render the appropriate shade of white, gray, or black for these values. Pure red at 100% saturation would have a value of 255, 0, 0. Pure saturated green would be 0, 255, 0. Likewise, pure saturated blue would be 0, 0, 255. All properly adjusted monitors should display the same photo with the same white point and color tones.
The problem is not that all monitors have their own white point. They can (and should) all be profiled to correct for variations. The problem is that not all light sources have the same white point. When we set the white balance on our camera we are telling it what color temperature the light is so that when that light is reflected by a white surface the camera and monitor can display the white objects in the scene correctly.
Let's assume for a moment you are saving your photos as RAW files. In this case the only effect the WB you set in camera has is on the JPEG thumbnail generated by the camera. This is because the data in the main file from each pixel well on the sensor has not yet been interpolated and adjusted for the varying sensitivities of the human eye to Red, Green, and Blue. When you open a RAW file using the appropriate software, the white balance setting in the program might be set by the program itself or the camera's setting when the picture was taken may be read from the file's metadata and applied. In either case you are free to select another. The effect of different WB settings is in how the 10, 12, or 14 bits per color in a RAW file are translated to the 8 bits per color needed to display on your screen.
I'm looking at a RAW file I shot recently under mixed tungsten and fluorescent lighting. A friend in the photo was wearing a gray shirt. By placing the mouse pointer over the shirt, the value of the pixel is displayed at the bottom of the screen. A spot with a value translated at 204, 162, 114 (orange-grey) when the WB is set at 5200K is translated to 163, 174, 185 (blueish-grey) when the WB is set to 3200K. At a setting of 3500K the grey item is correctly translated to 183, 183, 183. Assuming the grey shirt my friend is wearing is neutral grey, I just found my WB point.
So what happens if you are shooting JPEG? The camera assigns the WB you have chosen. If Auto WB is selected the camera assigns a WB of its own choosing based on the value of the brightest parts of the scene. It then translates the data coming off the sensor into 8 bits per color and discards the rest of the information before saving the file to your memory card.