"As a Kiwi-Coop we welcome competition, we're simply not allowed to share the details of our commercial arrangements," Clement said.
In the open letter to Fonterra chief executive Theo Spierings this week, Cullinane questioned whether Fonterra is playing fair.
He said he understood that "Fonterra is looking to use its market power to introduce an exclusionary deal with supermarkets in the North Island that would all but remove non-Fonterra brands".
Cullinane noted that the domestic market represents just a fraction of Fonterra's $20 billion business and calls on Spierings to be "supportive of the little guys" and not to crush innovation in the New Zealand market.
Leon Clement, managing director Fonterra Brands New Zealand, said: "Fonterra agrees that all Kiwis want to see fair play and have plenty of choices in the milk chiller. New Zealand consumers have, and will continue to have, a wide variety of choices when it comes to fresh dairy."
Fonterra was making significant investment in growing the premium milk segment to encourage more New Zealanders to enjoy premium milk, Clement said.
"Competition, including in the premium category, is critical to the success of the New Zealand dairy industry. It promotes efficiencies and innovative new products to meet public demand."
Cullinane also raised concerns about packaging similarities between Lewis Road Creamery products and new Fonterra brand products "which is likely to confuse shoppers".
Cullinane is a former advertising industry executive and a director on the board of NZME, which owns the New Zealand Herald.