No, Apple does not advise or recommend that you need or should fit a screen protector. Whether or not to fit a screen protector is entirely a matter of personal preference.
Do not use toothpaste or any other abrasive on your Apple Watch.
It is possible that you have only scratched through the fingerprint-resistant, oil repellent coating. It is normal for this coating to wear away under normal use, but it may have become scratched without damaging the screen beneath.
Unless the issue is due to a manufacturing defect, cosmetic damage is not covered under either Apple's Limited Warranty or AppleCare+. If your Apple Watch has more than cosmetic damage, then note that Apple advises that you should not continue using a damaged Apple Watch.
Apple Watch is not user-serviceable and Apple does not offer a screen repair or a crystal / screen replacement service for Apple Watch.
If the issue is not covered under warranty (or AppleCare+, if previously purchased), then your Apple Watch may be instead eligible for replacement via chargeable, out-of-warranty service.
If you wish to explore this option, then contact Apple Support (mail-in service may be available, if required), make a Genius Bar reservation or visit an Apple Authorised Service Provider to arrange for your Apple Watch to be inspected and to establish any available service options:
Note that, whether you have scratched the screen or not, note that Apple does not recommend using cleaning products or polishing or buffing the watch with abrasives; these will wear away the coating and may also (further) scratch the display.
A good quality screen protector (of the thin film type) may be effective in obscuring or reducing the visibility of the mark, as well as in providing some measure of protection against future damage.