  We prove the uniqueness, up to a pull-back by an element of a suitable
subgroup of complex automorphisms, of the weighted extremal K\"ahler metrics on
a compact K\"ahler manifold introduced in our previous work. This extends a
result by Berman--Berndtsson and Chen--Paun--Zeng in the extremal K\"ahler
case. Furthermore, we show that a weighted extremal K\"ahler metric is a global
minimum of a suitable weighted version of the modified Mabuchi energy. This
implies a suitable notion of weighted K-semistability of a K\"ahler manifold
admitting a weighted extremal K\"ahler metric.
