That's the problem, in a nutshell. Legislative precedent says that we draw the line at businesses refusing to serve demographic classes. Arguing to move that line is fine, but may lead to the slippery slope back to widespread discrimination based on race, religion, gender or any number of other characteristics that the civil rights movement spent decades fighting.
Provided that you have no problem with busineses refusing to serve Christians, then your argument is, at least, consistent. But, under current law, that would be illegal, and hence so is refusing to serve other groups, whether religious or sex/gender based.
And your point is arguably disingenuous at best. Aside from the fact that counting state outcomes, especially when not all states put it to a popular vote, does not permit any conclusion on the national level of popular support, virtually all polls in recent years indicate clear popular support for gay marriage. http://en.wikipedia.org/wiki/Public_opinion_of_same-sex_marriage_in_the_United_States
Are you asking how it is being claimed that religious beliefs, per se, are bigoted? Because that's not what is being claimed at all. Beliefs, religious or otherwise, are not necessarily bigoted, and not necessarily not bigoted. In this case, the rights being asserted, under the umbrella of religious freedom, appear to permit bigotry. That is the issue.
Partly inevitable - Fox is the only mainstream channel that is overtly right wing, so not surprising that it gets a lot of viewers. But another point that is often overlooked is that its basic news coverage is not that bad, and not the cause of its reputation for spreading nonsense. That's down to its talk shows.