Updated October 10, 2019: Tuesday's encore airing of CW's Batwoman series premiere was watched by 1.1 million viewers live, according to The Hollywood Reporter. The re-run ranked competitively with this year's The Flash season premiere which debuted Tuesday to 1.63 million viewers.
Between its two live broadcasts, Batwoman's season premiere was watched by an estimated 2.9 million viewers. The audience for delayed-viewing (streaming and DVRs, for examples) has not been made public.
Original Story from October 7, 2019: The CW's Batwoman debuted Sunday, October 6 with 1.8 million live plus same-day viewers, according to TVByTheNumbers. That would be the lowest Arrowverse series premiere to date, but the highest CW non-Arroweverse series premiere in 22 months - going back to January 2018, when the last Arrowverse show debuted: Black Lightning.
Those ratings placed Batwoman fifth in its 8pm timeslot, behind NBC's Sunday Night Football pre-game, Fox's The Simpsons, ABC's Kids Say the Darndest Things, and CBS' God Friended Me.
Supergirl, which returned for a new season directly after Batwoman, was watched by 1.3 million people and was second for its timeslot behind NBC's Saturday Night Football.
Compared to other Arrowverse debuts, Batwoman was relatively low - January 2018's Black Lightning series premiere earned 2.31m viewers live + same-day - but did not go up against an NFL game. Back in January 2016, DC's Legends of Tomorrow took in 3.21m viewers. Supergirl, which debuted on CBS, was watched by 12.96m people back in October 2015. The Flash was watched by 4.83m back in October 2015. Arrow debuted to 4.14m people in October 2012.
Batwoman is scheduled to air Sundays at 8pm on the CW.