In most countries (such as the USA) you do not need the proof of purchase to get warranty service.
Have you tried any troubleshooting or contacting Apple support? For example, you should at least try doing a forced restart (holding down the side-button and digital crown for 10+ seconds?) You might even want to try this several times, including while on the charger, and to try another charger if you know anyone who has an Apple Watch.
If you want to check the warranty status the serial number may be etched on the bottom of your Apple Watch Series 1, but if it is not it will be written inside where one of the bands attach in the hinge (you will have to remove the band to see it though). With that serial number you can check the warranty status here:
https://checkcoverage.apple.com
You should also be able to pull it up from the Watch app of your paired iPhone by tapping on the i-circle next to the picture of your Watch.