For the most part, courts have decided that the constitutional interest in providing equal access to public accommodations outweighs the individual liberties involved. The problem with saying no to doctors who might want to deny service but yes to a funeral home is you get into a raft of extenuating circumstances. That doctor might be one of thousands that can render the service. A funeral home may be the only one in town. One hospital might feel they should have the right to say no because there are a number of other hospitals in the town but maybe the patient insurance is not accepted elsewhere. My point is there is an infinite number extenuation circumstances and that creates a real problem with providing equal treatment under the law.
For most businesses denying service is bad policy. It can lead to boycotts, being vilified in the media, and loss of business Although gays and lesbians only constitute an estimated 10% of the population maybe less, gay people have mothers, fathers, siblings, friends, neighbors, and business associates. A business man that adopts a policy denying service to gays and lesbians may find he's taken a much bigger hit than he anticipated.
That's exactly what I've been saying. I don't this law would extend to any public services, like police, fire departments. I also THINK it's already against the law for a hospital to turn away a sick patient.
With your funeral home example, though, that would be an inconvenience to the gay person, that's pretty much about it. They might have to go to the next town over or something.
Keep in mind, I don't agree with bigotry and think that it's terrible, but I think it is a private business owner's right to do business with who he or she chooses. That doesn't make it right though. I think it's asinine just to be clear.

I just think the government has no right to say you MUST do business with this person if you own a small private business.