This packing problem is the limit as $s\to\infty$ of the problem of minimal energy point configurations under the Riesz potential $V=1/r^s$. Hardin and Saff show (see Theorem 2.1) that the minimum energy $E(A,N)$ of $N$ points on the $d$-dimensional manifold $A$ satisfies

$$\lim_{N\to\infty} \frac{E(A,N)}{N^{1+s/d}} = \frac{C_{s,d}}{\mathcal H_d(A)^{s/d}}\text,$$

where $C_{s,d}$ is a constant independent of $A$ and $\mathcal H_d$ is the $d$-dimensional Hausdorff measure. I think an appropriate limiting procedure can show that the optimal packing radius for $N$ spheres on $A$ satisfies a similar limit.

This packing problem is the limit as $s\to\infty$ of the problem of minimal energy point configurations under the Riesz potential $V=1/r^s$. Hardin and Saff show (see Theorem 2.1) that the minimum energy $E(A,N)$ of $N$ points on the $d$-dimensional manifold $A$ satisfies

$$\lim_{N\to\infty} \frac{E_s(A,N)}{N^{1+s/d}} = \frac{C_{s,d}}{\mathcal H_d(A)^{s/d}}\text,$$

where $C_{s,d}$ is a constant independent of $A$ and $\mathcal H_d$ is the $d$-dimensional Hausdorff measure. As $s\to\infty$, the energy is given by $E_s(A,N)=e^{-s (\log 2t(A,N)+o(1))}$, where $t(A,N)$ is the optimal packing radius of $N$ spheres on $A$. After a bit of work, you should be able to get the desired manifold-independent limit. It is also possible that the packing problem itself is treated in the literature separately, but I am only familiar with the cited reference.