Aer Lingus flight EI111, an Airbus A330 enroute Shannon to New York-JFK, had been cleared from FL 380 to FL 390. When climbing through FL 386, the crew reported encountering severe turbulence and requested and was cleared to FL 370. At the time of the occurrence, the A330 was flying 6.34 nm behind a United Airlines Boeing 777 at FL 390. The minimum radar separation required was 5 nm. There had been no previous reports of turbulence to ATC. The A330 was subsequently handed over to Boston ARTCC without further comment from the crew. Approximately 40 minutes after the turbulence encounter, Boston ARTCC advised Moncton ACC that the Aer Lingus flight had declared a medical emergency due to passenger injuries sustained during the turbulence encounter. One passenger suffered serious injuries, two suffered minor injuries.