Can the Apple Watch series 3 check your blood sugar by itself?

No, that would require some form of blood sample, which the Apple Watch is not capable of collecting (for now at least, thank goodness). Apple Watches can check your heart rate though, using pretty sophisticated technology: Your heart rate. What it means, and where on Apple Watch you’ll find it. - Apple Support

Hi

No current Apple Watch model offers glucose monitoring as a built-in feature.

Various third-party apps for iPhone - some of which include an app for Apple Watch - offer features that are designed to help with the recording and analysis of glucose levels, some when used in conjunction with specific third-party accessories.

For example:

One Drop includes an app for Apple Watch and can sync data directly from a Bluetooth-enabled blood glucose meter (available separately as a third-party accessory):

• One Drop for Diabetes Management on the App Store
• One Drop Blood Glucose Monitoring Kit - Apple

Dexcom G5TM Mobile Continuous Glucose Monitoring (CGM) System:

• http://www.dexcom.com/g5-mobile-cgm
• "View your glucose information, trend graph, and alerts and alarms right from your wrist with the Dexcom G5 Mobile Apple Watch app".
• Dexcom G5 Mobile on the App Store

Dexcom G4® PLATINUM System with Share

• http://www.dexcom.com/dexcom-g4-platinum-share
• "Dexcom Share2 is compatible with your Apple Watch. You can view your own glucose information and trend graph right on your wrist."
• Dexcom Share2 on the App Store

Keep in mind Apple's Important safety information, which includes:

Not a medical device Apple Watch, the heart rate sensor, and the included Apple Watch apps are not medical devices and are intended for fitness purposes only. They are not designed or intended for use in the diagnosis of disease or other conditions, or in the cure, mitigation, treatment, or prevention of disease.

Important safety information

If you would like to submit a feature request to Apple, then you can do so here:

Feedback - Apple Watch - Apple