Is there a legal argument to be made to include the right to privacy in the current dispute regarding first amendment rights and the government talking to socia

Is there a legal argument to be made to include the right to privacy in the current dispute regarding first amendment rights and the government talking to social media companies about removing information/accounts? Basically if the government can't undermine the 1st amendment through 'asking nicely' of companies then they also can't bypass the constitutional right to privacy? That would make it a broader constitutional rights coalition.