Democracy has nothing to do with giving or taking rights away from others. The "Public" have no right to tell another person who they can and cannot marry. What you are talking about is mob rule. Whereby the tyranny of the majority deny rights to a minority.
And what you're talking about is elitist rule, whereby the tyranny of the minority deny the rights to a majority.
As long as the US Constitution allows it (and, per the case of "Baker v. Nelson", defining marriage as one man and woman DOES NOT VIOLATE the constitution), the majority of the people DO have the final say.
And, yes, the "public" have the right to tell another person who they can and can't marry.
The "Public" says you can't marry more than one person at one time.
The "Public" determines the age that person has to be.
The "Public" says no marrying any relatives, closer than a first or second cousin.
And, for those who need to be told...
The "Public" says no marrying animals.