  Douglas and Rudin proved that any unimodular function on the unit circle $\T$
can be uniformly approximated by quotients of inner functions. We extend this
result to the operator-valued unimodular functions defined on the boundary of
the open unit ball of $\mathbb{C}^d$. Our proof technique combines the spectral
theorem for unitary operators with the Douglas-Rudin theorem in the scalar case
to bootstrap the result to the operator-valued case. This yields a new proof
and a significant generalization of Barclay's result [Proc. Lond. Math. Soc.
2009] on approximation of matrix-valued unimodular functions on $\T$.
