  A set $S\subseteq V$ of a graph $G=(V,E)$ is a dominating set if each vertex
has a neighbor in $S$ or belongs to $S$. Dominating Set is the problem of
deciding, given a graph $G$ and an integer $k\geq 1$, if $G$ has a dominating
set of size at most $k$. It is well known that this problem is
$\mathsf{NP}$-complete even for claw-free graphs. We give a complexity
dichotomy for Dominating Set for the class of claw-free graphs with diameter
$d$. We show that the problem is $\mathsf{NP}$-complete for every fixed $d\ge
3$ and polynomial time solvable for $d\le 2$. To prove the case $d=2$, we show
that Minimum Maximal Matching can be solved in polynomial time for $2K_2$-free
graphs.
