The Scotland midfielder was shown a straight red card for a kick at Neymar after bringing the Brazilian down during Tuesday night's Champions League clash at Celtic Park.
Celtic were comfortably holding Barca before the midfielder's 59th-minute dismissal and went on to lose 1-0 after Cesc Fabregas headed the winner with Celtic caught short of numbers at the back.
Brown, quoted on Celtic's official website, said: "I realise that I made a mistake in getting sent off last night. It's something that happened in the heat of the moment but I shouldn't have acted in this way.
"I still think it was a harsh decision. However, I recognise that I made a mistake and put myself and my team-mates in a difficult position.
"I need to live with that and there is no-one more disappointed than me about losing the match.
"There wasn't a huge amount of contact at all, to be honest, but I have to accept responsibility for my actions - and I do.
"The team deserved to get something out of the game after playing so brilliantly and putting in so much effort against one of the best sides in the world.
"I was also disappointed for the supporters after the incredible backing they gave us during the game. They were magnificent once again."
The defeat left Celtic without a point after starting their campaign with another narrow defeat, when a late deflected goal sent Milan on their way to a 2-0 win in the San Siro.
But Milan's draw with Ajax leaves the door open for Celtic to kick-start their qualification attempts as they prepare for a double header against the Dutch champions.
"It was another fantastic performance from the lads after the Milan game, and although we've still to pick up a point from this group, we've shown we can compete against some of the best sides in Europe," Brown added.
"We have four games to go in this group and everyone in the squad firmly believe that we can still qualify, and we are determined to get our first points on the board against Ajax in two weeks' time.
"For my part, I can't wait to get back on the pitch against Motherwell at the weekend and try and help us extend our winning run in the league."