Brown was given a straight red for flicking his boot at Neymar when the Barca forward went to ground during a challenge from the midfielder.
Celtic felt the punishment was excessive and Brown made a personal appearance at the governing body's headquarters in Switzerland to contest the decision.
Brown has already missed Celtic's 2-1 victory over Ajax and will now sit out the return fixture in Amsterdam next Wednesday and the home clash with AC Milan on 26 November.
The Scotland international ran the risk of having his ban increased, but the suspension stands at three games meaning he will be available to play against Barcelona at the Nou Camp in December.
Celtic are currently third in Group H above Ajax with three games remaining in the section.
Manager Neil Lennon told Sky Sports News "The appeal has been turned down. It's disappointing as we thought we had a good case.
"My own personal feeling is the ban is excessive for the misdemeanour, or whatever you want to call it, but I'm sure he had a fair hearing.
"Obviously it is a blow for us and a blow for Scott, but we just have to deal with it."