Unless a governmental entity is doing the censorship, the First Amendment does not apply. Here the TV network is free to take whatever steps they want and there is no violation of the First Amendment. This is basic constitutional law. Here is a brief explanation of this requirement from Cornell Law School http://www.law.cornell.edu/wex/state_action_requirement
The state action requirement stems from the fact that the constitutional amendments which protect individual rights (especially the Bill of Rights and the 14th Amendment) are mostly phrased as prohibitions against government action. For example, the First Amendment states that “[c]ongress shall make no law” infringing upon the freedoms of speech and religion. Because of this requirement, it is impossible for private parties (citizens or corporations) to violate these amendments, and all lawsuits alleging constitutional violations of this type must show how the government (state or federal) was responsible for the violation of their rights. This is referred to as the state action requirement.
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