The incident happened around midnight on Saturday in Buccleuch Terrace.
Police said the man was walking with a group of friends through the Meadows when they were approached by another group who began an unprovoked confrontation with them.
The 19-year-old and his friends attempted to leave and headed towards Buccleuch Street, into Buccleuch Terrace. The other group followed and continued their confrontation, causing the man to fall to the ground where he was kicked in the head.
The attackers fled the scene before police arrived.
The man was taken to Edinburgh Royal Infirmary for treatment to a suspected fractured eye socket.
A police spokesman said: "This was an unprovoked assault that has left one man with a particularly painful and serious injury to his eye socket.
"Anyone who was in either the Meadows or Buccleuch Terrace from around 11:55pm and has information that can help with our enquiries should contact police immediately."