Persistent rain forced an early abandonment of action on the clay courts in Houston, with all eight second-round matches pushed back to Thursday.
Max Purcell's clash with Daniel Altmaier was the only match to get underway, with the Australian leading 6-4 before rain intervened.
Top seed Frances Tiafoe was due to open his campaign against Steve Johnson, with second seed Tommy Paul to face Yannick Hanfmann.
Nakashima's withdrawal for an undisclosed reason meant lucky loser Zizou Bergs will replace him and take on Cristian Garin.
Djokovic is bidding to tie Roger Federer and Rafael Nadal with a record 20th grand slam championship by defending his 2019 title.
Since losing the opening set of the tournament to Jack Draper, Djokovic has been imperious form and he eased past Garin on Centre Court on Monday.
This was not vintage Djokovic, but his consistency on serve, strength on the return and remarkable movement skills helped him secure a 6-2 6-4 6-2 triumph to set up a last-eight meeting with Andrey Rublev or Marton Fucsovics.
Djokovic raced into a commanding position in the first set, winning the opening eight points of the match.
Indeed, Garin, aiming to reach the quarter-finals of a grand slam for the first time, appeared overmatched, winning only 10 points on serve as Djokovic took the first set with ease.
But the Chilean warmed to the occasion and displayed his resilience by saving three break points to hold for a 4-3 lead in a game that lasted over 10 minutes.
Garin's resistance was broken in his next service game through some deft Djokovic touch at the net and, after serving out the second, he broke in style in the first game of the third and raced through to a 50th grand slam quarter-final.
Data Slam: First serve fires Djokovic to half-century
Djokovic faced two break points in a one-sided contest, with Garin rarely able to threaten his first serve.
Indeed, Garin won 13 points on return and only three of those came against the Serbian's first serve in a performance he will surely want to forget.
By contrast, Djokovic won 48 per cent of points on Garin's serve and, save for the second set, rarely had to expend much energy in bringing up his half-century.
WINNERS/UNFORCED ERRORS
Djokovic – 28/23
Garin – 14/26
ACES/DOUBLE FAULTS
Djokovic – 9/1
Garin – 2/5
BREAK POINTS WON
Djokovic – 5/12
Garin – 0/2
The Australian, a beaten quarter-finalist at Wimbledon in 2014 and 2015's Australian Open, dominated en route to a 6-4 6-3 7-6 (7-5) win in the tournament's first quarter-final between two unseeded players since 2008 (Arnaud Clement v Rainer Schuttler).
Kyrgios has courted controversy on multiple occasions over the last fortnight, spitting in the direction of a fan in the opening round before winning a bad-tempered clash with Stefanos Tsitsipas in round three, but he put the theatrics to one side in clinching a last-four spot on No. 1 Court.
Garin started strongly as he broke to love in the contest's opening game, but Kyrgios, energised after producing one of the shots of the tournament when stooping for a scarcely believable forehand winner, broke back shortly after.
After overcoming a back-and-forth ninth game to record another break, Kyrgios served the opener out to love before moving through the gears in the second, which he wrapped up in 38 minutes to extend his lead.
Kyrgios's fine serving allowed him to take control of the contest, with his 17 aces in the match moving him clear of John Isner (114) as the player with the most aces at Wimbledon this year (120).
The Australian lost his way for much of the third set, producing a series of unforced errors to give Garin hope, but the Chilean's failure to convert any of his last eight break points saw Kyrgios force a tie-break.
Kyrgios rediscovered his form in the breaker to see out the win, setting up a last-four meeting with either Taylor Fritz or Rafael Nadal.
Data Slam: Kyrgios flying the flag after making quick work of Garin
Kyrgios' win ensured he became the first male Australian player to reach Wimbledon's semi-finals since Lleyton Hewitt in 2005.
Having being taken to five sets on two occasions in West London this year (by Paul Jubb and Brandon Nakashima), the 27-year-old enjoyed a more comfortable outing on Wednesday when extending the best major run of his career.
WINNERS/UNFORCED ERRORS
Garin 30/23
Kyrgios 35/29
ACES/DOUBLE FAULTS
Garin 2/5
Kyrgios 17/4
BREAK POINTS WON
Garin 1/9
Kyrgios 3/6
