This is just bold speculation: The spiral pattern you see arises from a homomorphism from the integers to the unit circle, and the irrationality of $\phi$ gives it a pleasant aperiodicity. In higher dimensions, there isn't an abelian group structure on spheres, so the naive generalization using $d$-parameter families of integers doesn't work. On the other hand, there are infinite discrete subgroups of orthogonal groups whose orbits have dense image in a sphere, so you can take the orbit of a point under the action, ordering by something like word length. An explicit example of such a free faithful action of $F_2$ on $S^2$ is given in David Speyer's blog post, using the matrices $\begin{pmatrix} 3/5 & 4/5 & 0 \\ -4/5 & 3/5 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 3/5 & 4/5 \\ 0 & -4/5 & 3/5 \end{pmatrix}$.
I have no idea what the resulting "spiral" will look like, or how "well-packed" it is.
Addendum: If you choose suitably random orthogonal matrices, you will still get a faithful action of $F_2$. You might have fewer near-collisions with a "less hyperbolic" group, but I don't know much about the geometry.
