I recommend using the excellent Hammer.js[0] library to pick up swipes (since that's the gesture you're looking for). I've put together a small proof-of-concept here[1]. Try it out on your phone.
Alternatively, if you wanted to have some slick dragging action (so that a user could swipe a little, and control the rate at which the menu drawer opens), you could put a smaller (maybe 10-20% width) dragbox along the left-hand-side of the screen, and keep your selectable content to the right of that.
[0] http://eightmedia.github.io/hammer.js/
[1] http://codepen.io/notduncansmith/full/sthyJ/
