Falloon resigned as MP for Rangitata in July.
The petition has received more than 39,000 signatures.
"It's time to put the health of young Kiwis first."
He thanked the party and board members for their "vote of confidence" in him.
"If we underestimate Jacinda and her advisers, we will be in Opposition for a very long time."
National Party president Peter Goodfellow has led a tirade against the Labour Government.
Three board positions are up for grabs including the presidency.
They reportedly went back despite being asked to stay away until they had the all-clear from health officials.
It is the third complaint upheld against the party this year.