Gillespie, who is a vegan, made the comments in an interview with the Yorkshire Post.
Wensleydale Creamery signed a two-year extension to its sponsorship of the Headingley side in April.
"I hope one day the dairy industry can be shut down. I think it's disgusting and wrong on so many levels," he said.
The former Australia pace bowler added: "It's out of my control, just like the fact that cricket balls are made of leather.
"I'll have it out with people, I don't care. There's nothing wrong with standing up for what you believe in."
A spokesman for the sponsor said it would seek more clarity on Gillespie's comments.
"That's a very bold statement, and I need to understand a bit more about what he's actually trying to say there," he said.
"We are very proud sponsors. Our milk comes from over 40 local farms with very high standards of welfare."
Yorkshire coach Jason Gillespie has risked upsetting one of the club's main sponsors after saying he "hopes one day the dairy industry can be shut down".