The maverick Australian toppled Tsitispas in a heated four-set thriller on Saturday, during which he called for the fourth seed to be kicked out of the tournament for hitting a ball into the crowd.
Kyrgios called the umpire a “disgrace” as tempers frayed and the bad feeling spilled over into the post-match press conferences.
The defeated Greek player said his opponent has an “evil side” and described him as a “bully”, comments that Kyrgios laughed off.
Kyrgios, 40th in the world, thrives on his bad-boy image and Saturday’s outbursts were not even his first of the tournament.
But it will be a different challenge against American Brandon Nakashima, a player he has never faced before, and he may struggle to re-create the big-match intensity on Centre Court.
Kyrgios, who has only reached the quarter-finals of two Grand Slams, believes he has the firepower to win Wimbledon.
“Round by round, if I keep doing my things, I feel good. I’m all right,” said the 27-year-old, who had vocal support from the crowd on Saturday despite his antics.
Nadal, chasing the third leg of a calendar Grand Slam, found his rhythm in his third-round match against Italy’s Lorenzo Sonego and now faces Dutch 21st seed Botic van de Zandschulp.
The 22-time Grand Slam champion, twice a winner at the All England Club, swept into the last 16 in straight sets after needing four sets to progress in his opening two matches.
“My best match, without a doubt, since the tournament started,” said Nadal, the only top-10 player left in his side of the draw.
“I made improvements today. Very happy for that.
“I made a lot of things much better than the previous days, the determination, the way that I manage to play more aggressive, going to the net plenty of times.”
In the other two last-16 matches, Australia’s Alex De Minaur plays Cristian Garin of Chile while Australia’s Jason Kubler takes on 11th-seeded American Taylor Fritz.
Tan needed more than three hours to see off Williams but on Saturday she hit top form, demolishing Britain’s Katie Boulter 6-1, 6-1 in just 51 minutes.
Tan, ranked 115 in the world, said she was enjoying Wimbledon.
“I think I like grass,” she said. “I never play on this court but I like to play with slice and volley and everything with my game so I’m really happy.”
The most eye-catching match of the day in the top half of the draw is a contest between fourth seed Paula Badosa of Spain and 2019 champion Simona Halep, who is yet to drop a set.
Romania’s Halep, who is the 16th seed this year, said becoming a champion at the All England Club gave her confidence.
Badosa is the only member of the world’s top 10 — along with Tunisia’s Ons Jabeur — left in the women’s tournament.
French veteran Alize Cornet, who ended Polish top seed Iga Swiatek‘s 37-match winning run on Saturday, faces Australian Ajla Tomljanovic while 17th seed Elena Rybakina of Kazakhstan goes up against unseeded Croatian Petra Martic.