Can Western Blots be used to quantify the activity of a protein?
I see 3 parts to a complete answer:

If you define the phosphorylated protein as active, and non-phosphorylated protein as inactive, and you have a total protein blot (as you say, via pan-specific antibodies) as you describe, then you have done everything correctly.

It's simple math from there on based on radioactivity or fluorescence or mass or whatever you use to quantify your band intensity. The 3 components should match up with the inactive protein amount (active + inactive = total). If this is consistent, you have additional evidence that you have done it correctly.

However, here come the hidden assumptions: this is all assuming that (i) you are not experiencing significant protein loss during your preparation steps, that (ii) your antibodies are good (specific and yield the complete protein extract), that (iii) the blotting procedure is performed correctly, and most importantly, that (iv) your initial definition is valid, that the phosphorylated protein is truly the active form of the protein, and that the unphosphorylated counterpart is the inactive form. You see, this is not always the case, though with some (most?) kinases, it is. If it has not previously been demonstrated, one may raise the reasonable objection that your assumption is wrong. However, I take it that in your specific case, the kinase activation by phosphorylation is well understood.