A set S⊆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≥1, if G has a dominating set of size at most k. It is well known that this problem is NP-complete even for claw-free graphs. We give a (almost) complexity dichotomy for Dominating Set for the class of claw-free graphs with diameter d. We show that the problem is NP-complete for every fixed d≥3 and polynomial-time solvable for d≤2. To prove the case d=2, we show that Minimum Maximal Matching can be solved in polynomial-time for 2K2-free graphs.