Fica, claiming strong evidence that captains had been pressured by their boards, notably the powerful BCCI, to change their initial votes sought an investigation.
South African captain Graeme Smith was reportedly bemused at being asked to vote three times.
Frustrated by the ICC's tardiness in investigating the vote, Fica has referred it to the ICC ethics officer, Sean Cleary, for independent investigation.
Having met the ICC in London, Fica upped the ante, claiming "poor governance practices".
The statement signed by the New Zealand players says they "expect cricket's administrators to ensure our great game is managed off the field to the same high standards of dignity and integrity as those to which we are committed". The ICC responded angrily, claiming Fica was guilty of a "breach of trust of the processes and protocols agreed at their London meeting".
Fica executive chairman Paul Marsh said yesterday that Fica had tried without success for more than six weeks to have the ICC act. "However ... we are left with no choice but to refer the matter ourselves."
"Sadly this is yet another example of the poor governance practices that exist in cricket. The ICC should be taking these allegations incredibly seriously but instead they are ignoring the processes under their own code and hoping the matter will go away," Marsh said.
Quoting three clauses of the ICC's Code of Ethics in arguing its position, Marsh added that "it is extremely concerning and disappointing that the governing body of our sport has refused to follow its own processes for dealing with allegations of unethical behaviour".
The ICC hit back, claiming Marsh's statement was "confrontational" "emotive", and breached protocols agreed by the parties in London.
It pointed out Marsh notified the ICC of Fica's intentions at 1.31am UK time and put out its "misleading" statement at 4.30am on Thursday (UK time) when ICC board members, in London for their annual conference, were asleep.
"These actions do not reflect the spirit in which ICC and, we believed, FICA entered into what appeared to be meaningful and productive dialogue, nor reflect a willingness to work together to provide a satisfactory conclusion to this issue," the ICC said.
There is no indication how long Cleary's investigation would take, but if Fica's assertions are found to have substance it could throw an embarrassing light on the ICC's directors, who include New Zealand Cricket's chairman Chris Moller and ICC president, New Zealand's Alan Isaac.