Back in May, Kyrgios described Nadal as "super-salty", accusing him of being a sore loser and adding that his uncle and long-time coach Toni is an "idiot".
Following his first-round win over Jordan Thompson earlier on Tuesday, Kyrgios said he and Nadal have a "mutual respect", adding he was not sure they "could go down to the Dog & Fox [nearby Wimbledon pub] and have a beer together".
Nadal responded after his win by saying he is "too old" to get involved in a war of words: "[I've had] too many years on the Tour. I am not going to be in a fight with nobody. I'm here to keep doing the thing that I love."
Nadal, chasing his third Wimbledon title and first since 2010, had vented his frustration at his seeding going into the tournament, being displaced by Roger Federer in second.
But the Spaniard is on a semi-final collision course with Federer regardless after being drawn in the Swiss’ half, and on Tuesday he looked far more assured then he did at the Hurlingham Club last week when losing to Marin Cilic.
There was an early blip when Nadal dropped service in the opening game, but the 33-year-old recovered with two breaks of his own to take the opening set.
The second set was far more routine as Nadal broke three times without reply.
World number 274 Sugita gave a better account of himself in the third set, but his resolve was broken in the eighth game, and Nadal went on to close it out a game after.
Earlier on Tuesday, Kyrgios battled into the second round after a rollercoaster five-setter with fellow Australian Thompson.
After edging a mammoth third-set tie-break 12-10 to lead two sets to one, Kyrgios then lost the fourth set to love in just 18 minutes.
However, he bounced back to break twice in the decider and secure a 7-6(4) 3-6 7-6(10) 0-6 6-1 win.