Edema or Oedema? Which One is the Correct Spelling?

Both are correct.

From Twitter:

KidneyNotes Just overheard someone use the word "oedema" -- and they pronounced the "o." Oawesome. about 7 hours ago from Tweetie

BiteTheDust @KidneyNotes "use the word "oedema" -- and they pronounced the "o." Oawesome." perhaps they came from an English speaking country? ;-) about 6 hours ago from TweetDeck in reply to KidneyNotes

DrGourmet @AllergyNotes I always thought it was a contraction for "Oh! I have edema." 2 minutes ago from web in reply to AllergyNotes

Edema (American English) or Oedema (British English), formerly known as "dropsy" or "hydropsy," is an abnormal accumulation of fluid beneath the skin, or in one or more cavities of the body. Source: Wikipedia.

Technically, the Greek origin of the word requires "oe-" for correct spelling. There are many medical terms that differ between American and British English, for example: anaemia, oesophagus, haemorrhage, haemophilia, etc.