"So happy Australia Day. It is a special day for you especially," Courier said.
"Yesterday you found out that you were the Young Australian of the Year. Congratulations. How did you find out about it and what does it mean to you?"
Barty responded without any sign of discomfort: "Yeah, it was a surprise. Pat (Rafter) actually surprised me a couple of days ago, presenting me the trophy which is really special.
"It is extremely humbling."
The comment caused a stir on social media after Day 7 of the Australian Open was played out to the backdrop of nearby Australia Day protests in Federation Square, Melbourne.
Thousands of people also rallied at Parliament House in Melbourne to mark Invasion Day — to counter the celebration of the arrival of the First Fleet in 1788. Indigenous speakers addressed the crowd and told those assembled January 26 was a day of mourning.
It's why Courier's comment was seen as "awkward" by some.
Barty, however, delivered a message of absolute class in explaining why she was not in any way offended by Courier's remark.
"I'm an extremely proud indigenous woman. It's not for me to decide what day Australia Day is on," she said in her post-match press conference.
"Like I said, I'm staying true to my values. I am extremely proud of my heritage. That's going to remain the same every single day of every single year for the rest of my life. It doesn't really matter what day it is.
"I'm a proud, indigenous woman. I'm a proud Australian. For me, it doesn't matter what day of the year it is. I'm going to stay true to my values all throughout the year, all throughout my life.
"Every singe day I want to be kind, I want to be honest, I want to be humble, I want to give the best that I can. That's all that matters for me."
Barty, from Ipswich in Queensland, is fighting to become Australia's first Open singles champion on home soil since 1978.
She has been inspirational as she climbed the ranks to become the world's No. 1 women's singles player, as well as serving as the National Indigenous Tennis Ambassador for Tennis Australia.
Barty also made history on Sunday night after defeating rival Alison Riske in three sets.
The previous Aussie to qualify for consecutive Australian Open quarter finals was Pat Rafter in 1988.
In fact, no Australian has ever twice qualified for the singles quarter finals since the Grand Slam tournament moved to Melbourne Park in 1988.
