Sure it does!
The reason we say it happens with monochromatic light is the formulas work well with monochromatic light. The interference bands are dependent on the wavelength of light passing through the slits. If you have a range of wavelengths passing through the slits, you will have a range of interference patterns, and the formulas you use would have a third variable.
Rainbows are similar to white light interference patterns. In a rainbow, white light passes through droplets of water acting as prisms, splitting the light into it's component colors. When white light passes through a slit, the interference patterns are all shifted slightly depending on the individual wavelength considered to be passing through.
Lets say you have a slit some distance from a wall, and lights are passing through it. Say you shine a red light through, and on the far wall you get a space 1 cm wide between the bands. Then, you swap the red bulb in your light with a green one, and repeat the experiment. Now, the bands are a different distance apart, say 1.07 cm. What do you think would happen if you shone both at the same time? You would see the two wavelengths produce their two spectral patterns. White light is just a wide range of visible lights, so this theory would extend there.