No it does not, the government is not the entity they are dealing with.
Either the principle is important and you support it or you don't. If you actually believe in freedom of speech you don't get to celebrate censorship by corporate entities because the constitution didn't forbid it.
This all-or-nothing view has never been true, and that's kind of the point. Freedom of speech requires social dialogue and consequence to have value - the ability of private entities to reject what they don't like is what makes the whole thing work.
I believe that Google should feel free, both legally and socially, to act according to their conscience as a company. However, I believe that their conscience, as a massively powerful part of online communication, should be to remain neutral towards any legal communication that takes place on their platform. I think that's a more effective moral stance than the one they are taking.
People cheering Google here, were you happy when Occupy Wall Street was run out of a public park, essentially censoring the movement? I certainly wasn't.
However, Zucotti Park is private, not public.
Do we want to support a world where corporations can freely censor ideas they don't like? Is that a good world to live in?
The text of the first amendment is: Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.
It says nothing about private entities regulating private platforms and that's on purpose - free speech means nothing without the private means to accept or reject it.
I feel that most of this confusion stems from people who live in different countries conflating their free speech laws with our own. In the United States, the government cannot regulate speech on the basis of its content, and this has always been the case. In the United States, private entities are free to regulate speech on private platforms, and this has also been the case since day 1 of the republic. This is nothing new for us Americans, and nothing particularly scary!
In other words, because it isn't constitutionally protected doesn't mean I have to like it, and I certainly won't give them kudos for it. I'm not arguing the legality of it, you are and I'm not sure why. I suspect it's because you support this type of censorship.
Can you even think of one government protected right we've gained in the last 20 years? Don't say pot, because we lost that right in the 1900s. As a citizen it is your duty to be diligent to protect your freedoms, for your self and future generations. The road to hell is paved with good intentions, and we're well down it.
My solution was to have a public option registrar.
As long as companies are making individual decisions and ICANN isn't involved, I think it's good that companies are making a moral stand. But when we start creating blacklists we become no better than the Nazis. It's a line we're getting dangerously close to and we don't want to cross it. I see no problem with a company like Google banning neo-Nazi websites. I see a huge problem if all the registrars conspire to create a blacklist or if ICANN starts censoring content.