To some extent, no. In my opinion* you shouldn't get to say "I refuse to design websites for gay people."

*: Not always the same as Sam Alito's

That's not usually the problem (it can be for some). It's really when your art/"identity" is being used to promote something you don't believe in. That happens across the whole religious and political spectrum... e.g. musicians that don't want their art used in certain contexts.

About the OP, government is the right place where we "fight it out" and try to sloppily design a system to move forward as one; these seemingly activist campaigns to "jawbone" private companies is absolutely a sign that something needs to be done at the government side. However, there are very entitled, rich interests behind the banking system, so yeah, there isn't one easy solution i think.