"Bad times poor result and racist abuse from a minority of fans," he wrote, adding in a follow-up tweet: "no problem with a bit of banter but a line was crossed today."
The 33-year-old, who has been with the New Zealand-based Phoenix since 2009, said it was the first time he had experienced such treatment in the A-League.
Wellington's Dominion Post newspaper said police and Adelaide club officials were informed of the abuse allegations and Football Federation Australia was investigating.