The red-hot Draper has yet to drop a set or a service game at the tournament and dominated his Czech opponent 6-3 6-1 6-2 for the 22-year-old's deepest run at a Grand Slam tournament.
"Incredibly proud of that achievement," Draper told reporters. "To follow in Andy's footsteps and make the quarter-finals here since he did it in 2016, it's a big achievement for me.
"But I think I've just got to keep on going because I know that there's still room for improvement and still room to hopefully go further in the tournament."
Murray, who won the U.S. Open in 2012 for the first of his three major titles, retired after the Paris Olympics and while Draper said the 37-year-old is always just a text away, he did not plan on reaching out to him at the moment.
"Probably not," he said.
"Obviously I feel like he would completely understand and respond to any messages I send him and would give me advice, but I think also at the same time I think he's kind of enjoying his retirement now and he's kind of done with it."
Draper beat Murray in their only ATP Tour level meeting at Indian Wells last year in a match that felt at the time like a changing of the guard for British tennis.
"Andy I obviously miss," Draper said in an on-court interview.
"Shout out to him. What an unbelievable career the guy has had. Icon of the game. I miss him in the changing rooms. I miss being next to his stinky shoes and all of his stinky clothes," he said.
"Andy is a legend. If I have half the career that he's had, then I'll be a happy man."
