A biomechanical analysis can be done to determine how much support comes from the muscles vs other (skeletal) support structures. Assume the body is standing upright, in static equilibrium. Consider this (over simplified) free body diagram of the upper torso where the sum of the forces equals zero in static equilibrium.
Here the diagram over simplifies the analysis with one general equation: Fsupport = sum of all the weights. Since we are considering only how much weight muscle supports, we deduct Wbones from Fsupport.
A spine by itself would not be able to support itself because the sum of the forces do not equal zero in static equilibrium. (Wgravity + Wspine) > Fsupport.
In reality, each muscle, bone, and organ has many more forces with specific magnitudes in different directions that altogether equal the weight of the specific region of mass it supports. I imagine these forces as many vectors over an area/field. A more detailed analysis would require a program like MATLAB that uses triple integrals in calculus to visualize these forces in 3D. This kind of visualization can help us color code only the muscle vectors which answers your question, "to what extent is our ability to support our spine to remain upright due to muscles," as a percentage of the sum of all the forces input into the program. How the muscles keep the spine erect can be determined from such visualizations as descriptions of directional forces.
For an example of how this kind of analysis is done, see this paper on lumbar spinal support. http://www.uvm.edu/~istokes/pdfs/stab-model.pdf