Assuming you have a way to connect to the internet via your apple watch like via a carrier or wi-fi connection you should be able to do so. If you just try putting your iPhone into airplane mode and emailing yourself from another device or the watch you should be able to test this theory.
However I am not sure about other providers that are not iCloud as your watch should have a secure token to your iCloud mail so I would think you can still send and receive there but not sure about other providers and how the watch will access those mailboxes.
But now I am coming to try it I am having little luck but would assume better success if the iPhone is online somewhere even if tis not close to the watch as it appears all mail is bridged with the iPhone.