Do American ministers have the legal right to decline to marry gay couples?Do American ministers have the legal right to decline to marry gay couples?
Ministers are free to not marry somebody of another faith. This is pretty common.
A church can decide it will only marry a couple behind closed doors where the non-faithful can't enter. This isn't so common, but may happen. I believe that Mormon weddings that are performed in their temples don't let in non-Mormons.
Someone can decide to not marry a couple because they're of the wrong color, or of mixed colors. Yes, this still happens in the 21st century. It's uncommon enough that it makes news sometimes when it happens, but it still happens. (One example, but certainly not the only one: Interracial Couple Spurned).
A minister can decide not to marry something if they don't agree to whatever type of counseling the church favors, or if they don't make a donation of the appropriate size, if they don't agree to a set of values the church espouses, or if they don't agree to certain wedding vows or locations.
Gay marriage opponents have ignored the facts and made up a reason not to allow gay marriage out of whole cloth.
ETA: Congratulations to those of you who were looking forward to being allowed to marry and now can. I'm very happy for you.