
Bublik won the first set 7-5 and was up 2-0 in the second when Sinner pulled out with a right leg injury.
The Italian had taken a medical timeout at the end of the first set.
He received treatment from a physiotherapist and played, and lost, two more games before pulling out.
Sinner, the 21-year-old world No 9, now faces a race against time to be fit for Wimbledon which starts in early July.
The match was the opening last eight tie.
In later action, home favourite Alexander Zverev takes on Chilean Nicolas Jarry.
No 1 seed Daniil Medvedev plays Spaniard Roberto Bautista Agut while third seed Andrey Rublev faces Dutchman Tallon Griekspoor.