A lot of people conflate that situation with First Amendment rights. It actually has very little to do with the First Amendment, rather it should be covered under the Civil Rights Act.
The First Amendment only stops the government from restricting speech and expression; the government can't force the bakery nor the gay customer to say or not say a specific message. Likewise, if the bakery wants to refuse service to someone, they can't claim 1A protection as a cover for that refusal because it simply doesn't apply. The 1A allows them to not be charged and prosecuted for speaking out against homosexuality, but that's it.
However, under the Civil Rights Act, they cannot refuse service to a protected class. Where it gets murky is that homosexuality is not explicitly defined as a protected class, only sex (male or female at the time of enactment). Today though, we are learning that "male" and "female" are not the only recognized sexes and there are non-binary definitions coming out all the time. The question becomes, is homosexuality a different sex as it pertains to the CRA?
I don't know the answer to that question. The way I see it though, if you are proclaiming to be a Christian yet you use that proclamation to discriminate against a particular class or individual, you are going against the teachings of Christ. According to Christian lore, he went out of his way to help the lowest of the low in his society, even those who were considered immoral or sinful by that society (prostitutes, lepers, thieves, etc). I'm not overtly religious (I was raised Baptist but I reject a lot of that sect's philosophies), but when I hand spare change to a homeless person on the street, I don't ask for sexual orientation before doing so. I see someone in need of assistance and I help where I can, because who they are doesn't matter, it's what they need that matters.
By the same token, I don't feel a business owner should care what someone's lifestyle or orientation is no matter their own beliefs, but in this nation it seems the Religious Right feels empowered to use their take on religion as a crutch to be cruel and dismissive towards those who are not perfectly in line with their own beliefs. Yes, they have a right to refuse service in certain cases, but when doing so steps on another person's right to even exist as they are, I feel it crosses a line that should not be crossed.
To put it another way: a gay couple's existence is not an aggression towards a homophobic person; there is no harm intended nor is there harm done by simply existing. But, the homophobe's negative actions towards that gay couple is an aggression.