Yes, you can!
Certain aspects of sleep tracking and detection will function better if Pillow is installed on the Apple Watch, therefore, it is strongly recommended to install Pillow on the Apple Watch. However, Pillow can still detect sleep without having Pillow installed on the Apple Watch.
To track sleep automatically, simply wear your Apple Watch during sleep and Pillow will produce a detailed sleep analysis that includes:
Detailed sleep stage analysis
Heart rate analysis
Blood oxygen levels during sleep (on supported devices)
Environmental audio levels (on supported devices)
Respiratory rate levels during sleep (on supported devices)
Wrist temperature tracking (on supported devices)