Yes, this is a Constitutional issue

I read several conservatives who support Phil Robertson, but don't think private employers discriminating against Christians violates the First Amendment. I disagree, and here's the source of their mistake.

To begin with, you have some conservative libertarians who think businesses have the right to hire or fire whoever they please. On a related note, conservatives who don't think this is a First Amendment issue are basing that on their own understanding of the First Amendment. Judging by original intent, the free speech, free exercise, and free assembly clauses only apply to the public sector, not the private sector.

Now, there's nothing intrinsically wrong with these arguments. In fact, I think they are right in principle.

Problem is, that's a paper theory. That's not how our current system actually works. In reality, state and Federal gov't has interjected itself into the private sector. For instance, there are laws which prohibit discrimination based on real or perceived sexual orientation, transgender identity or self-image.  "Hate" crimes (including "hate" speech) are another case in point. 

To say the First Amendment doesn't apply to cases like Phil Robertson would only be true if both sides were playing by the same rules (i.e. original intent, libertarianism). Since, however, the state is coercing private business on these very issues, it would be an act of unilateral disarmament for conservatives to play the game by a different set of rules. You will lose every time. 

It is, of course, worthwhile to challenge the current status quo. But as long as that's the operative framework, you can't have one side play by the rules while the other side is free to break the rules or unilaterally make new rules which overrule your rules. 

If, in a football game, your team plays by the rules while the other team has a portable goalpost which moves closer when its own team has the ball, but further away when the opposing team has the ball, that's not a fair fight.