He was ruled out of the bout with Russia's David Ayrapetyan because of a fractured nose.
Irish head coach Billy Walsh confirmed that following a meeting with Barnes on Friday that they had decided to withdraw the Belfast man.
"The safety of our boxers is more important than gold medals", said Walsh.
Barnes' nose was bleeding in Friday's semi-final victory over Salman Alizada of Azerbaijan and he needed attention from the ringside doctor during the bout.
"Out of the final today! Couldn't bluff the doctors today again, had to get an X-ray and nose is broke! Broke it in my 1st fight! #raging," Barnes tweeted on Saturday.
Ayrapetyan received the 49kg gold medal on a walkover.
Barnes, a gold medal winner at the 2010 European Championships, secured silver.
Meanwhile Michael Conlan also took silver after British boxer Andrew Selby successfully defended his amateur title in a split decision win.
John Joe Nevin and Jason Quigley were crowned European Champions after they won their bantamweight and middleweight finals.
Sports Minister Carál Ní Chuilín congratulated the Belfast boxers on winning silver medals at the European Boxing Championships.
"Throughout these Championships Michael and Paddy have demonstrated why they are outstanding athletes," she said.
"Yet again they have taken local boxing to an international stage proving that they are ambassadors for their sport and their communities.
"Congratulations to all of the squad even those who have not reached the finals, it has been a tremendous week of boxing and I am pleased that the tradition of boxing success on this island continues to grow from strength to strength."