Novak Djokovic, Rafael Nadal, Roger Federer, Naomi Osaka and Serena Williams were among those skipping the tournament that was delayed 2 1/2 years because of the coronavirus.
Fritz rallied from a 5-2 deficit in the third set by winning four of five games - when Zverev held two match points - to force the tiebreaker.
Angelique Kerber, the 2019 runner-up and highest remaining seed at No.10, lost to 21st-seeded Paula Badosa 6-4, 7-5.
In women's quarterfinals, former world No. 1 Victoria Azarenka beat American Jessica Pegula 6-4, 6-2 and will face Jelena Ostapenko in the semis.
Ostapenko rallied from a break down in both sets on a day when the weather calmed down from the rain and wind that affected some of Monday's matches.
Medvedev, the US Open champion, connected on 70 per cent of his first serves and fired four aces despite challenging weather conditions.
Clijsters, a 38-year-old Hall of Famer on the comeback trail, also lost in the first round at the WTA Tour stop in Chicago last week.
American Shelby Rogers, who upset top-ranked Ash Barty in the third round of the US Open, was set to play Anhelina Kalinina of Ukraine in a night match.
Emma Raducanu stunned the tennis world earlier this month by becoming the first qualifier in the professional era to win a Grand Slam singles title.
The withdrawal was announced by the official Twitter feed for the hard-court tournament at Indian Wells, which is scheduled for Oct.4-17 after twice being delayed during the coronavirus pandemic.
The tournament will feature prize money of $15.3 million, organizers said Thursday.