The role of GPS in INS/GPS navigation systems Ideally, a gyroscope and an accelerometer would be enough for a complete navigation solution (attitude + position), using dead reckoning. This comprise the Inertial Navigation System, INS. In non-ideal world, we would couple the INS with GPS. The GPS helps 'fix' the navigation solution at certain times, to help mitigate the drift in the INS output.

My question.
Suppose I have no access to a GPS, but have access to a Spirit level instrument with associated camera and image processing algorithms. This system serves to provide an absolute values of pitch and roll angles, but with slower update rate and lower resolution. Still, it is assumed to be drift-free. Can such a system be used in place of normal GPS to achieve dead reckoning solution?

P.S. The reason for not using GPS is that I am trying to find a self-supporting navigation solution...
By itself, no, it cannot.

Your method of fixing the INS system is pretty clever, but it only accounts for drift in the gyroscope. Accelerometers are also (very) prone to drift, generally showing up as an increasing velocity bias. The typical method to account for accelerometer drift is to use zero-velocity updates. By detecting points when you are not moving, you can periodically remove any non-zero velocities from your model.