Pospisil, with just two main-draw wins this year going into the tournament, was up 6-4, 2-0 when eighth-seeded Jurgen Melzer was unable to continue. It was unclear why Melzer retired.
Pospisil did lose his serve twice but had four breaks in Melzer's six games serving.
He moves into a second-round match against Michael Yani, who was a 6-3, 7-6 (7-4) winner over Brian Baker. Yani recorded the only break of the first set and survived losing his serve twice in the second set by taking the tiebreaker.
Wild-card entrant Jimmy Wang ousted Riccardo Ghedin 6-3, 6-3 behind a nine-ace attack. Wang next faces second-seeded Juan Monaco, who had a first-round bye.
Igor Sijsling was a 6-2, 6-1 winner over Ariez Elyaas Deen Heshaam, a local favorite, in Monday's other match. Sijsling was on the court just 50 minutes and won 72 percent of the points on serve and 62 percent when receiving.
Selena Gomez drops F-bomb, walks off stage during Jingle Ball performance
Wisconsin business offering 'therapeutic cuddling' forced to close