Howard said he has been given no indication as to why his candidacy was blocked by a reported six of the 10 major cricketing nations at the ICC's meeting in Singapore.
"Even in private discussions they are very reluctant to give a reason," Howard told Australia's Sky News.
The ICC asked both Cricket Australia (CA) and New Zealand Cricket (NZC), who nominated Howard to the role, to resubmit a new candidate by August 31. The vice president will become the president of the governing body in 2012.
"I won't be withdrawing," Howard said.
"I wanted to do this job, I wanted to do it well and I would have devoted my full time to it."
Howard had been a contentious choice for the position, with Australian media reporting that his criticism of Zimbabwe president Robert Mugabe's regime while Australian Prime Minister had alienated him with both South Africa and Zimbabwe.
"All I can say in relation to Robert Mugabe is that if that is the reason I wear the negative attitude as a badge of honour because I don't apologise in any way for the criticism I offer of the Mugabe regime," Howard said.
"If it was in some way based on past political reasons then that is a very bad precedent to be establishing for the ICC because there are serving politicians holding positions of authority within the ICC.
"I'm not criticising that but I'm just drawing attention to it."
CA and NZC said on Wednesday they are "deeply disappointed" with the decision.
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