"I had the best dad in the world," he captioned the video.
"Thinking of him and remembering nothing but fun times.
"I hope you all had a wonderful Father's Day in the US - sending love to those who couldn't celebrate with their Dad."
Father's Day is celebrated on September 4 in New Zealand and Australia.
The Australia Zoo star also took to his Instagram account with a throwback photo of the pair.
He wrote a similar post for the caption, "Happy Fathers Day ❤️ I am thinking of my Dad today and everyone out there who might not get to celebrate with their's. It's so important to me to remember and cherish all the special times. Nothing but fun!"
The 18-year-old's sister commented, "Love you so much".
On Twitter, Robert posted another tribute - this time praising his brother-in-law Chandler for being a great father.
"Father's Day is a day filled with emotion - I think of my Dad a lot & remember him fondly," Robert wrote.
"But now, I also think about & admire my brother-in-law Chandler & what a fantastic father he is to my niece, Grace.
"I'm happy she has such a fun, kind person to help guide her through life."
Robert's followers sent touching messages of support after watching the clip.
"I was not emotionally prepared for this tbh," one said.
Another wrote: "He was a dad to anyone who didn't have a father to take them on adventures as a kid & taught some of us how to be brave when encountering wild animals."
"He would be so proud of you Robert Irwin and Bindi Irwin," another replied, adding a heart emoji. ️
