You are never going to convince a bigot to change his (or her) mind by pointing out that their belief is bigotry.
You may, at the cost of becoming unruly or rude yourself (which will likely lose you some potential allies), make it so unpleasant for the bigot to voice their bigotry in public that they shut up about it.
Public silence will not stop them from being bigotted, though & they may well take their views underground, only to re-emerge stronger later. Something of that sort has happened with intolerant religious beliefs in general in America, which were driven underground in the Sixties & have come back roaring in the past decade or so.
We can also see this in the religious pushback over the Chick-Fil-A situation & over other matters. To these folks, voicing their bigotry is a religious freedom. As we have freedom of speech here (for however much longer that lasts), everyone has the right to speak their opinions, even those who opinions you or I may find disgraceful (literally, not full of grace). Objecting to free speech is not a liberal virtue, so when liberals do it, they can rightly be called hypocrites & other names.
The only real cure for bigotry is for members of the group being belittled by it joining the bigots’ circle of friends, business associates, fellow students, church members and families. If you want to truly end bigotry, you have to be there, both to show the bigot that they are wrong about you and to keep them from poisoning the next generation with their ignorant beliefs.
This is why desegration in education is so important & why keeping your freak flag in the closet when you are around mundanes is essential.