NoThe following would be a counterexample if we require only $v_i \ne v_j$ but don't forbid different vertices to become antipodes on the sphere. Let us call this a weakly faithful orthogonal representation.
The graph of the octahedron has a weakly faithful representation in (real) dimension 2, given by $\pm e_i$. If you remove one edge, there will be no faithful representation in dimension 2: the remaining edges force the two non-connected vertices to lie at distance $\pi/2$ on the sphere.
This example seems to generalize to higher dimensionsSo, for faithful orthogonal representations the question is: can one force the distance (with the graph ofin the cross-polytopestandard spherical metric). between two points in the projective space to be $\pi/2$ by imposing distances $\pi/2$ between some pairs of points?