The 24-year-old, who has emerged as one of the game's more volatile characters this season, reacted badly to being handed a game penalty for unsportsmanlike conduct in the final set of his last-32 showdown with Norway's Ruud, kicking the ground in frustration, slamming his racket down before throwing his wooden chair into the middle of the court.
Furious at the decision, Kyrgios booted a water bottle and then stunned spectators by picking up a fold-up wooden chair and hurling it across the court before picking up his bag and storming off, although he did shake hands with Ruud and the chair umpire.
He added that while Djokovic was an "unbelievable player", there was no way the "cringeworthy" player would surpass the talented Roger Federer.
Kyrgios branded Spanish great Nadal a "super salty" sore loser who was partly responsible for turning fans against him.
Eight-time Rome champion Nadal cruised past Jeremy Chardy 6-0, 6-1 and will next meet Nikoloz Basilashvili, while Djokovic eliminated Denis Shapovalov 6-1, 6-3 and will play either Philipp Kohlschreiber or Marco Ceccinato.
All three are in action again later in the day to make up time after play was washed out on Wednesday.
For 37-year-old Federer, returning to Italy for the first time since 2016, it was like a practice session. "Do this for me, do this for me, I won't say hello to you, I'm too important".
Another outburst at the fan ensued: "I've been giving 100 percent and I have to deal with f--g idiots like you. I believe it's going to be similar conditions in the match this afternoon".
"Good thing was this one was not that physical", Federer said.
"The celebration just kills me".
"I'm quite pissed about it".
He said: "The first thing that comes to mind is the greatest of all time, for me".
Japan's Kei Nishikori, the sixth seed, got past American Taylor Fritz 6-2, 6-4 and next meets Jan-Lennard Struff.
"He just wants to be like Roger", Kyrgios told the No Challenges Remaining podcast with Ben Rothenberg when asked what he thinks of world number one Djokovic. Rudd was declared the victor and now faces Juan Martin del Potro in the third round of the Italian Open.
Halep's defeat also guarantees Osaka's top seed at the French Open.
Federer fell behind 40-0 in the next game but then won five straight points to secure a 5-3 lead, closing out the service game with a forehand cross-court passing shot on the run from the corner - drawing a loud roar from the Foro Italico crowd.
Czech second seed Petra Kvitova eased past Yulia Putintseva of Kazakhstan 6-0, 6-1.