Supposing it was New York Time, sure something quite unimaginable, but supposing this web host decides to terminate their contract and no other host in the country would provide them with a hosting service, would that be an infringement of freedom of speech?

Perhaps if we view it from the perspective of the black letter of the law, that is the constitution, the highest law of that country, it might not be an infringement. However freedom of speech goes beyond the constitution, it is a principle ingrained in our society and culture.

To address your differences between private and public, a newspaper of course has no obligation to give him space in their platform, nor the television, but the internet is different. The former two are selective, the internet is free for all. If private bodies are able to go around policing the kind of content that is to be found on the net, the difference between them and the government is a mere technicality.