It's possible to measure the heart rate optically at the wrist. A few consumer products already exist:
For 24/7 monitoring: http://www.mybasis.com/
For sports: http://www.mioglobal.com/ and http://micoach.adidas.com/
The Basis comes quite close to what you suggest.
There's much more information in a heartbeat than just the rhythm [0]. I don't know what exactly the Basis can record, and whether more information would be useful for automatic diagnoses at all. For example, some Garmin and Suunto watches also measure the time between heartbeats to estimate the "training effect" of a workout. You should ask a cardiologist what kind of data is required to detect certain abnormalities.
Also, you will probably get a lot of false positives. If you search for "heart rate monitor spikes" you'll see what I mean.