In a word, no.
For a full explanation of what the Constitutional amendment regarding freedom of speech does and does not protect, read: Freedom of Speech - Let Me Tell You What It's Not
In a nutshell, the 1st Amendment (like the rest of the Bill of Rights) places limitations on governmental action...and governmental action only.
There is a lot of confusion over this issue, and people on message boards often complain that moderators who remove their comments (or even other participants on the message board who simply complain about others' comments) are somehow interfering with their free speech rights. Unless the moderator or other member is acting under color of law, as a representative of some branch of government, that's simply not possible.
With very few limited exceptions, the private owner of a website, blog, forum, etc. can set any limits he or she likes, and delete any comment he or she likes.
