You can, but only using the Apple Music app. I've asked Google Play music repeatedly to have something similar on the watch, but they don't seem to care. Like you, I wanted to have ONLY my watch while running (I like to minimize any external clothing/devices while doing so).
So what I do now is use an armband as a place to keep my iPhone while running or exercising. The watch DOES let you seek forward/back once you have any music app playing (this works for podcasts as well) so effectively, I use the watch as a convenient "controller" for my iPhone.
Protip if you're oncall/devops: I LOVE the watch when running because I can quickly acknowledge pages from PagerDuty since you can scribble and send text right from the watch itself. Once ack'd, I finish my run and get back to handling the page