Global Peace and Justice Auckland urged chess player Victor Mikhalevski to make the "sacrifice" and sit out the tournament, which began yesterday.
Mr Minto, a GPJA spokesman, wrote to Mr Mikhalevski, saying the events in Gaza were an "outrage".
In a statement, Mr Minto and fellow GPJA spokesman Mike Treen said if Mr Mikhalevski did not withdraw, they had asked New Zealand Chess Federation president and tournament organiser Paul Spiller to request he step down.
Mr Minto said the letter had been sent a day before the tournament began but no response had been received from Mr Mikhalevski or Mr Spiller.
Mr Spiller and Mr Mikhalevski did not want to comment, the latter saying he preferred to focus on the tournament.
Mr Minto, a major figure in the 1981 Springboks tour protest, called on Israeli tennis player Shahar Peer to withdraw from the ASB Classic tournament in Auckland earlier this month.
She declined to do so.
