Freedom to Marry is financing a $1 million campaign to boost support for same-sex marriage.
Utah is appealing a federal judge's December ruling.
The state fails to make its case for a temporary ban on gay marriages, attorneys representing gay couples argued Friday.
Utah's attorney general has asked the Supreme Court to halt same-sex marriage.
The state is expected to file a request for a temporary gay-marriage ban any day now.
With so many cases before federal judges, it is likely that 2014 will be the year in which states accounting for a majority of Americans will allow gay marriage.
Ruling applies only to death certificates, but it is expected to be precedent-setting, leading to more lawsuits in Ohio challenging the law.
A federal judge ruled Utah's ban on same-sex marriages unconstitutional Friday, writing in his opinion that it "perpetuates inequality."
Ten others recognize unions and partnerships while 33 limit marriages to opposite-sex couples.
Some 2.77 million U.S. users participated in the late-March pro-gay marriage campaign.