Spain won all seven games in Germany en route to lifting the trophy, but England gave them a hard fight in the final, losing 2-1 to a Mikel Oyarzabal goal four minutes from time.
"As always in these games, it's fine margins. But I think they were the best team in the tournament and over the piece they deserved to win," Southgate said.
"We've competed until the very end of the final. I think today we didn't keep possession of the ball quite well enough," Southgate said after Spain outplayed England for long periods and enjoyed 63% possession.
"They press you really well and you've got to get out of that pressure and we weren't able to do that and in the end that meant they had more control of the game, and that can wear you down a bit.
"That said, we were still right in there when we get the equaliser, so the game was still wide open," he said, ruing a late chance for England when a header was cleared off the line.
Southgate, desperate to give England a first trophy since their 1966 World Cup win, did not comment on his future.
"It's going to take a while to pick all the bones out of it really," he said, acknowledging the sub-par performance of captain Harry Kane who lacked sharpness and was substituted.
"Physically, it's been a tough period for him. He came in short of games and not quite got up to the level we'd have all hoped," Southgate said.
Kane was equally devastated.
"Losing in a final is as tough as it gets. We did really well to get back into the game, to get to 1-1. We could have used that momentum to push on. We couldn't quite keep the ball and we got punished for it," he said.
"It's as painful as it could be in a football match, It's the last stage of the tournament. There's a lot of tired legs, a lot of tired mentality there, It's a huge disappointment."
Kane said the players wanted Southgate to stay.
"We've made it clear we love the manger, that's his decision, this is not the time to talk about that now. That's down to him. He'll go away and think about it. Right now we're all just hurting," he said.
