"I won't be withdrawing," he told Sky News late on Wednesday.
"Even in private discussions they are very reluctant to give a particular reason," he added. "It's a very unusual situation."
The snub was greeted with anger in Australia, where former International Cricket Council (ICC) boss Speed called the move "a disgrace and an insult" to Australia and New Zealand, which nominated Howard.
England was the only other country to back the appointment, giving the row an unfortunate racial tinge.
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Cricket Australia (CA) chairman Jack Clarke said he was "gutted" by the decision, adding it "was in the grand final" of ICC slights towards the country.
"(I'm) gutted and incredibly disappointed that a man of John Howard's stature has been knocked off for this job," he said.
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