  Large-scale {\em ab initio} QED calculations are performed for the
$2p_{3/2}$--$2p_{1/2}$ fine-structure interval of Li-like ions with nuclear
charges $Z = 5\,$--$\,92$. Improved theoretical predictions are obtained by
combining together two complementary theoretical methods, namely, the approach
that accounts for all orders in the binding nuclear strength and the
nonrelativistic QED approach that accounts for all orders in the
nonrelativistic electron-electron interaction. The resulting unified approach
provides theoretical predictions which are more accurate than the available
experimental results across the interval of the nuclear charges considered.
