The Celtic captain went face to face with the Portuguese head coach at half-time after Caixinha became incensed over a perceived elbow by Brown on Rangers striker Alfredo Morelos.
Caixinha said after the match that he told Brown “loud and clear” what he thought about the coming together.
After the match, Brown took to social media to taunt the Ibrox boss, posting a photo of the confrontation on Instagram.
He captioned the picture with: “When you should just listen and not speak. #loudandclear”
Brown also made fun of the Rangers boss on a similar post by Mikael Lustig. The Celtic defender posted an image of Caixinha going to ground after being inadvertently slide-tackled by Josh Windass.
On that picture, Brown commented: “That’s a dive.”