Kyrgios defeated the Russian the last time they met, less than a month ago in Montreal, and their meeting was marked as one of the highlight probable matches at the hardcourt Grand Slam since the draws were announced.
"I want to be playing on the biggest stadiums in the world in front of millions of people broadcasted globally," the temperamental Australian said.
"That's where I want to be. That is why I work hard. It makes it even sweeter. I know a lot of people don't want me to succeed. I know a lot of people hate my game, the way I do things, the way I go about it."
Kyrgios harnessed his mighty serve to clinch his third round match in an efficient hour and 56 minutes, whacking 21 aces at his opponent and sending over 35 winners, as he wrested the momentum early with a break in the fifth game of the first set.
There were few fireworks as Kyrgios defended all seven breaks he faced inside Louis Armstrong Stadium, where he earned a $7,500 fine for "spitting and audible obscenities" during his second-round meeting with France's Benjamin Bonzi.
With the sporting world's eyes laser-focused on neighbouring showcourt, as Williams fell to Ajla Tomljanovic in dramatic fashion in what is widely expected to be her final tournament, Kyrgios' lopsided win prompted little fanfare.
But it nonetheless marked the latest in a stellar run for Kyrgios, who is playing the best tennis of his career after winning in Washington, D.C., and reaching his first Grand Slam final at the All England Club.
Kyrgios said he knew exactly the way he was going to play against Medvedev.
"There's no shame in losing to a player like that. But the way I'm playing, the game plan and the success I had against him obviously in Montreal, I definitely feel like I have a fighting chance," Kyrgios said.
"Not many players can say that right now, going in to play Medvedev at the US Open.
"We've got major respect for each other. I've got major respect for his coach as well. He's a great guy. Been really nice to me. Look, I think it's going to be a fun battle."
