Bautista Agut, the No. 18 seed from Spain, won 6-3, 6-4, 6-0 under the lights at Louis Armstrong Stadium.
Murray was initially scheduled to face Kyrgios but the Aussie withdrew just before the start of their match, throwing Sunday's schedule into chaos.
At Wimbledon last week, Kyrgios had to retire after the second set of his third-round match against Felix Auger-Aliassime because of a strained abdominal muscle.
The pair teamed up for the first time and defeated Austin Krajicek and Sabrina Santamaria 6-3, 3-6, 7-5 on a packed Court Two.
The controversial Australian crowd-pleaser reached the Wimbledon third round by brushing aside Italy's Gianluca Mager 7-6 (9/7), 6-4, 6-4 to set up a clash with Canada's Felix Auger-Aliassime.
Williams, the seven-time Wimbledon champion, slipped and hurt her heavily-strapped right leg before quitting her first round tie after just six games against Aliaksandra Sasnovich.
His was one of more than two dozen first-round contests that wound up finishing on Day 3 after rain Monday and Tuesday scrambled the schedule.
The Australian crowd pleaser unleashed his rapid fire service on the back of four successive aces in the first game of the fourth set against French 21st seed Ugo Humbert.
Alexander Zverev, playing for the third day in succession, was shocked by Ernests Gulbis, ranked 138, in a gruelling five sets.
Kyrgios has been in trouble for his immature behaviour on and off the court several times in his short career.
Andy Murray won 6-1, 6-3 in the south coast English town of Eastbourne in Monday during a warm-up event for Wimbledon.
Murray was playing only his second match after 11 months on the sidelines with a hip injury that required surgery in January.
World number 88 Karlovic blasted 18 aces to defeat the 22-year-old Aussie fourth seed 3-6, 6-2, 6-3 in 92 minutes.
The 6-4, 6-2 victory over the American gave a fourth title for the enigmatic 22-year-old Australian, and the first since 2016.
Kyrgios, dubbed a 'human headline' by Australian media, is coming off his best year yet when he won three titles.