@Robert: This was too long for the comment field and so I'm putting it here.
I have another example which backs up what you're saying. Hermitian symmetric spaces of compact type admit a K"{a}hler embedding into complex projective space. If we consider two points $x$ and $y$ on the embedded submanifold $M$ then there are two distances; the geodesic distance $d_M(x,y)$ in the submanifold and the Fubini-Study distance $d_{FS}(x,y)$ in the projective space. An implication of your result is that it is not possible to express $d_{FS}(x,y)$ as a function of $d_M(x,y)$ (except for complex space forms). I looked at the case of the complex Grassmannian for which there are explicit formulas for geodesic distance in terms of stationary angles. It's clear that you are correct; a simple calculation shows that it's not possible to express the Fubini-Study distance solely as a function of the geodesic distance. Thanks very much for your answer.