We study the existence and computation of spherical rational quartic curves that interpolate Hermite data on a sphere, i.e. two distinct endpoints and tangent vectors at the two points. It is shown that spherical rational quartic curves interpolating such data always exist, and that the family of these curves has n degrees of freedom for any given Hermite data on Sn, n ≥2. A method is presented for generating all spherical rational quartic curves on Sn interpolating given Hermite data.