Can you get a new screen on your Apple Watch

Go to this page to find out more about the Watch repair service.: Apple Watch Service & Repair - Apple Support

For cost estimates go to the "Get an Estimate" menu, select the service type, then your series model, then the specific sub-model you have.

Typically, Apple doesn't replace parts on Apple Watch. For a price though, they'll take your watch and you pay the fee and get a refurbished, but like new Apple Watch in return of the same model as you handed in. This all depends of course if your watch is still eligible for service. Your tag line says you own Apple Watch Series 1, which is now too old to qualify for service options.

If you do own a Series 1 watch, it would far smarter monetarily to recycle the watch and buy a newer, more modern Apple Watch. It just doesn't make sense to invest any money in a watch which is no longer supported and was released nearly 8 years ago.