It does't currently measure breathing rate, but it may soon with iOS 14 - https://www.theverge.com/2020/3/9/21171483/apple-watch-ios-14-code-blood-oxygen-features-update . Next Apple Watch will supposedly have Pulse Oximeter to detect blood oxygen levels, it already has electrocardiogram (EKG/ECG) using these it will be able to measure the respiratory rate. Interestingly, Apple had filed a patent for this in 2017 - https://www.patentlyapple.com/patently-apple/2017/06/apple-invents-an-apple-watch-feature-to-measure-respiration-rate-with-multi-band-plethysmography.html .
There are non-contact ways to measure respiratory rate as well, such as using web camera as explained in this paper - https://www.hindawi.com/journals/js/2018/4567213/ . This can be useful for those who have issues with contact based mechanisms, such as myself; traditional smart watch/health band based heart rate monitors hurt me - https://abishekmuthian.com/my-experience-with-fitbit-charge-hr-numbness-tingling-and-pain-fec85d41d165/ .
So even when Apple brings respiratory rate detection with the new watch, I think there will be need gap for non-contact based respiratory rate detection at consumer level.
So even when Apple brings respiratory rate detection with the new watch, I think there will be need gap for non-contact based respiratory rate detection at consumer level.