Yes, I am sure Fred Phelps voted Obama, the pro-gay rights Democrat President.

I'm probably still registered as a Christian somewhere, doesn't make it so.
Well, I shouldn't be surprised Fox would throw that one out, it was a matter of time. But then, you see this kind of thing from conservative sites.

I would by no means suggest that being Republican and being racist are mutual, but I would suggest that today many Republicans are bigots. There were republicans who have supported the rights for black people and of course, Abraham Lincoln himself was a republican. However, the principles of being a republican or a democrat are not based on civil rights issues or being conservative or liberal or being left or right wing. Essentially I would suggest the difference between Republican and Democrat is how the country is run, though arguably both parties rule as representative (rather than direct) democracies, which is what a republic is. But it seems in the US, 'Republican' has become synonymous with other traits and likewise being a democrat has become synonymous with others. Unfortunately Republicanism in the US has become very corrupt.
Today civil rights is more of a liberal democrat issue than a republican one, not to suggest democrats are the only ones who are concerned about civil rights. You just have the religious right representing the Republicans and many are bigoted and naturally feel that as the bible condemns things like homosexuality that it should be law. It's interesting that it was Thomas Jefferson, another republican, who talked about the separation of church and state, not something you'd expect a republican to say today. And of course the ideals of the Westboro Baptist Church are much closer to the religious right than liberals. If anything, the ideals of republicanism are against the social elite and aristocracy, and yet we have the republican party favouring the rich and making the Church elite.
So arguably, the terms 'republican' and 'democrat' have lost their meanings. The above and Fox's attacks are just petty attempts to get a one up. To be fair, there's liberals who do the same, but I think it's stupid to promote this level of ignorance. But hey, if you encouraged rational thinking and intelligence, then it's a lot harder to manipulate people's opinions.
But Fox are just feeding the crazy, because the crazy brings ratings. That's all the Murdock mafia cares about, you're talking about the same network that airs Family Guy and stands against everything it represents on its news channels.