The out-of-sorts Croatian lost in straight sets to Queen's Club debutant Diego Schwartzman.
Argentinian Schwartzman, 26, secured early breaks in both sets to wrap up a 6-4 6-4 victory, only his third career win on grass.
British interest, in the singles at least, came to an end when Kyle Edmund succumbed to top seed Stefanos Tsitsipas.
The British number one was a set down and 3-3 in the second when rain caused the first-round match to be suspended on Wednesday evening.
Edmund gamely fended off six match points but Tsitsipas got lucky with the seventh, Hawk Eye showing his backhand had just clipped the line to secure a 6-3 7-5 win.
Greek youngster Tsitsipas will be back on court later in the afternoon to play his second-round match against Nicolas Mahut.
Second seed Kevin Anderson, who beat Britain's Cameron Norrie in round one, was sent packing in three sets by France's Gilles Simon.
The big-serving South African, last year's Wimbledon runner-up, has missed a chunk of the season through injury and it told as Simon ran out a 6-1 4-6 6-4 winner.
French qualifier Nicolas Mahut was given a warning after hitting a ball girl in the face with a stray serve during his second-round win over Stan Wawrinka.
Mahut, having just lost a service game, sent a practice serve back over the net, inadvertantly hitting the young girl.
She was reduced to tears and, after being comforted by Wawrinka, had to leave the court.
Mahut received a code violation for unsportsmanlike conduct from umpire Mohamed Lahyani, but BBC TV commentator John Lloyd felt the punishment was unnecessary, saying: "It was a complete accident. I don't think you really need to give a warning for that. I hope he doesn't get a fine."
It was the second time Mahut has accounted for a ball girl in two matches, having accidentally slid into another youngster during his first-round win over Frances Tiafoe.
Mahut won the match 3-6 7-5 7-6 (2) to set up an all-French quarter-final with Simon.
