It's true that there are infinitely many circles on a sphere through a given point with a given initial velocity. (Just intersect the sphere with any affine plane that contains the initial point and to which the initial velocity vector is tangent.) But if you want a great circle, then there's only one, namely the intersection of the sphere with the linear subspace of $\mathbb R^3$ spanned by the initial point and the initial velocity, regarded as vectors in $\mathbb R^3$. It's given by a simple formula.
Suppose $p$ is a point on the sphere and $v$ is a vector tangent to the sphere at $p$. (Here I'm thinking of both $p$ and $v$ as elements of $\mathbb R^3$.) Let $a = \|v\|/\|p\|$. The great circle with initial point $p$ and initial velocity $v$ is parametrized by $c(t) = (\cos at)p + \frac{1}{a}(\sin at) v.$ If the sphere has unit radius and $v$ is a unit vector, then this simplifies to $c(t) = (\cos t)p + (\sin t)v.$