Using my apple watch to listen to Spotify

To be clear only Apple Music can stream to the watch currently. You can download Apple Music to your Watch. Spotify can only be played via your iPhone. You can use your Watch as a remote control. Read this to help you further

https://support.apple.com/en-us/HT204691

watchOS 5 doesn't currently support direct streaming to the watch.

ASC - Late 2012 Mac mini, macOS 10.14.6; Watch, series 3 cellular, watchOS 5.3; iPad Air 2 & iPhone 8+, iOS 12.4; TV 4th Gen, tvOS 12.4;  HomePods, audioOS 12.4 and Airport Express w/ Airplay 2

I'm not sure, because I know that there are limitations when streaming from Spotify with the Watch. You may wish to check with Spotify support to answer that. I'm guessing that you have a cellular watch, or are you saying that you downloaded music to the phone, or to the watch?

Again, it is better asked of Spotify. I know there are a number of restrictions on what you can do without the iPhone nearby with the Apple Watch, but there are also a number of restrictions as to what you can do with Spotify on the Apple Watch. I'm not a user, so I cannot give you any better information than that.

That is "Today's" status on streaming. We do not know what will happen this Fall when watchOS 6 is released, and whether Spotify will provide an Apple Watch streaming app. If Spotify is your preferred streaming service, you should really check with them for their future plans.

But regardless what any company says will be coming in the future, you must always take it with a "Grain of Salt" as things can happen to cause promises not to happen as the future becomes today.

Yes, you currently need your iPhone. You cannot stream 3rd party music to the watch directly with watchOS 5x.