Democratic council vice president John Barth says Indiana needs to present itself as a welcoming place for all people. All 15 council Democrats supported the resolution, along with seven Republicans. Six Republicans voted against it.
Republican councilwoman Ginny Cain said she opposed the resolution because the marriage amendment's fate should be decided by voters.
Mayor Greg Ballard signed the resolution last week.
Republican legislative leaders say the General Assembly will vote next year on whether to put the amendment on the November 2014 statewide ballot.