Ashley
For your games, basically you need to make sure you draw backgrounds a little wider (or taller, depending on orientation) than the window size, to ensure no gaps ever appear.
So I guess using this method you should make your backgrounds no smaller than 4096 x 2160 if you plan on running it full screen on a PC as well to cover every possible resolution? The Radeon HD 7970 currently has a max resolution of 4096 x 2160.
I had been using a method to stretch the canvas size with html to fill the screen but there were certain issues with it. Since then I've been using one of the full screen options to scale the game and then a few conditions to fill the screen to eliminate the black bars without changing my graphics. Same approach as a TV uses to eliminate the black bars.