“He is a player with huge talent and could be winning Grand Slams or fighting for the number one ranking,” said Nadal.
“But he lacks respect for the public, the opponent and for himself.”
Kyrgios, ranked 72 in the world, hit back at Nadal.
“I’m different, Rafa’s different. He can focus on what he needs to do. He doesn’t know the journey I have been on, he doesn’t know anything about me,” said Kyrgios who famously stunned Nadal as a teenager at Wimbledon in 2014.
“I am not going to listen at all. That’s the way I play. The way he plays is very slow between points.
“I’ve got my game, he’s got his game. People are different, that’s the sport.”
Nadal, playing his first tournament since falling to Novak Djokovic in the Australian Open final, had three chances to close out Wednesday’s match after taking a 6-3 lead in the third-set tiebreaker.
Kyrgios saved the first with a leaping drop shot, and the second with a net-skimming volley.
Nadal was just wide with a passing shot to make it 6-6, double-faulted to give Kyrgios a chance and sailed a backhand long to end the match after three hours and three minutes.
Kyrgios screamed and dropped to the court in celebration — a conclusion that seemed unlikely when he complained to a trainer after the opening set that he was feeling ill but feared the fallout should he retire.
Kyrgios steeled himself to continue, and distractions seemed forgotten as he became absorbed in the battle.
Down 0-40 in the ninth game of the second set, Kyrgios saved four break points to keep the set on serve and raced to a 5-0 lead in the tiebreaker.
Kyrgios, who received treatment on his lower back before the start of the third, was in trouble again in the sixth game, but fought off five break points to stay on terms and push it to the decisive tiebreaker.
Kyrgios leveled his head-to-head record with Nadal at 3-3 and will play three-time Grand Slam champion Stan Wawrinka for a place in the semi-finals.
Switzerland’s Wawrinka, seeded third, belted 32 winners in a 7-6 (7/5), 6-4 victory over American Steve Johnson.
Women’s top seed Sloane Stephens also bowed out Wednesday, stunned by Brazilian qualifier Beatriz Haddad Maia 6-3, 6-3.
Haddad Maia’s victory over the fourth-ranked American was her first over a top-10 player in seven opportunities.
“It was just a tough match, obviously she played well,” said Stephens, who is playing in Acapulco for the first time since winning the 2016 title. “A tough day. But I’m not too sad about it.
“I’m just going to go back and work some more, practice some more and get ready for Indian Wells.”
Haddad Maia, 22, next faces China’s Want Yafan, who was leading Monica Puig 4-1 when the Puerto Rican retired with an injury.
Former world number one Victoria Azarenka of Belarus sailed into the quarter-finals, downing Tatjana Maria 6-2, 6-1.
Azarenka fired 20 winners to Maria’s 10, and took full advantage of the German’s 28 unforced errors.