Galaxy lost on 5-4 penalties to Real Salt Lake after the game ended 1-1 following extra-time.
"We all want to win titles and personally I'd love to be successful," England midfielder Beckham told reporters.
"But I think we have been successful this year without winning tonight, we have quietened a few people along the way which is always nice but we couldn't finish it off."
"Of course, I said I would be back after hopefully the World Cup and that's not going to change," he said.
"I had three injections but they wore off after 50 minutes," he said.
"Bone bruising is like that, one of those injuries that doesn't really go away until you rest it and take some time off. So I am just going to have to that in next few weeks," said the midfielder who heads to Italy at the end of December.
"I'll just rest now. It is the end of the season and I have to just try and rest it and get ready. I'll have rehabilitation because it has been like this for a few weeks," he said.
"I've got past that, whether people want to question my commitment, nothing like that is going to make me lose any sleep. If people still want to do it, let them do it," he said.
'He's a class act, he really is, you know. I wouldn't expect that from anybody and for him to come in here and just congratulate us... he's just a class act, top drawer and I mean he's done a lot for our league.
"I know it wasn't easy for him to lose this game. He wanted it more than, just as much as everybody else. I'm sure it was tough for him but he came in here and congratulated us and that was big time.'
"It was a great atmosphere, great city to have a final in. There was a buzz around the city, we knew it would be a good atmosphere and it was."
