Apple Watch 3 and using fall detection without having iPhone

Fall Detection is NOT available in the Series 3

Fall Detection was introduced in the Series 4 (which has been discontinued), and current exists in the Series 5.

The GPS only Series 4 and 5 do support Fall detection, and will make the emergency call if the iPhone is nearby or if the WiFi calling is supported by the iPhone's carrier and enabled on the iPhone and Apple Watch, and the Apple Watch happens to be on WiFi.

The Series 5 Cellular+GPS model has additional emergency calling abilities

Series 5 Fall Detection and Emergency Services on Cellular Apple Watch Series 5 with no cellular contract

https://support.apple.com/HT206983

https://www.apple.com/watchos/feature-availability/#branded-international-emergency

https://support.apple.com/HT208944

It will only work without the phone if you have the cellular version of the watch,if you have the GPS only version of the watch then you have to have the phone with you.