Uber got rid of their Apple Watch app, so you can no longer do this. You're limited by Apple Watch apps created by the vendors.
Could you go out without your phone? It depends. Can't take photos while you're out since there is no camera. Can't order an Uber while you're out since there is no app. Can't modify a spreadsheet on your watch, again no app. The battery life is also far more limited compare to a phone. You could spend an hour on the phone without killing the battery but an hour on the phone through the watch will hammer it badly.
I don't take my phone with me when I run, but that means I can't do those things. I can still make/take a call, send/receive texts, listen to music etc., but it's designed to be a companion to the phone, not a replacement for a phone.
So yes, you could go out without the phone, but it has limits.