Hampered by a hip injury, the young Australian went down to fourth seed Medvedev 6-4 6-3 6-2 in their third round clash on Rod Laver Arena.
Despite battling through pain and calling for medical assistance multiple times, the 20-year-old said he could not bring himself to retire.
"I just couldn't pull the pin. My first time on Rod Laver, I just wanted to dig it out and fight," Popyrin told reporters on Saturday night.
"I kept telling myself to fight throughout the match, even with injury."
Popyrin said he was unsure how serious the hip issue was, but his Open run proved last year's third-round effort "was no fluke".
He also made the third round of the US Open in September.
"I think I perform really, really well in grand slams. Hopefully I can keep performing like that," Popyrin said.
"I reckon I pushed him (Medvedev) a little bit on his serve towards the end of the first set.
"Then the second set, he kind of grabbed his game and started playing really, really well."
Medvedev will take on three-time grand slam champion Stan Wawrinka for a place in the quarter-finals.
He has a perfect 2-0 record against the Swiss veteran, with victories at last year's US Open and in 2017 at Wimbledon.
"A very tough opponent for a fourth round. Stan always plays good in majors, even when he's not in his best shape," Medvedev said.
While his third-round victory seemed comfortable, the 23-year-old said it was far from that.
"It was a great match, especially playing against Alexei - it's not easy," Medvedev said.
"To be honest, I have no expectations (for the tournament).
"Every time I lose a match, I'm super disappointed. I just try to win as many matches as I can."