Not really. Type II generally leads to either: antibody-receptor interactions (Graves) or antibody-antigen reactions (normal ADCC, warm hemolytic anemia).
Both IgG and IgM can fix complement, so in a normal response (type II) a foreign bug can be killed by either IgG or IgM in a type II response.
Type III is more commonly associated with complement, since IgM pentamers settle on vessels after binding antigen, where complement cleans them up (often resulting in vascular destruction). While this is what we more commonly associate with complement and immune complexes, we can also see complement fixation in type II.
More commonly, in type II autoimmune conditions, complement is not fixed, as any circulating antibody-antigen pairs are cleared by the spleen.
As far as the claim that Graves disease leads to complement fixation or phagocytosis, I would have to say that is false.
