Possibly, but would likely require quite a work-around as the gyroscope is gravity-based and has more physical attributes. You would either need a sort of "reference point" for the app to use like a light source, but then you run into image recognition technology. I'm not much of a programmer, but if I was limited to just a camera, I would have the user calibrate the app first or at least the room they're currently in. It would go something like this:
Center your head, and keep it as stable and forwards as possible.
Now, look up when prompted on screen.
Now rotate your body 180° clockwise, or counter-clockwise.
Room calibration is now complete!
The issue you have with this however, would be that this would differ if the room changes, or if the user moves to another room. Something of this magnitude would be akin to taking a panoramic photo like you stated, but instead using the images as a reference for the app to know where the user is looking. This is quite a predicament. I would also take into account the range of quality in mobile cameras, as this will indefinitely have a substantial impact on performance. I also advise including a sort of disclaimer telling the user that their devices camera may not be up-to-par, or limit the apps availability to higher-end devices. Your best alternative would probably be an accelerometer, but it sounds like your aim is for a more common piece of hardware. Hope this is helps.