The full quote does not support your position. The limitation on public authority is included in the right, but it is not the limit of the right.
Article 10 – Freedom of expression
1. Everyone has the right to freedom of expression. This right shall include freedom to hold opinions and to receive and impart information and ideas without interference by public authority and regardless of frontiers. This article shall not prevent States from requiring the licensing of broadcasting, television or cinema enterprises.
2. The exercise of these freedoms, since it carries with it duties and responsibilities, may be subject to such formalities, conditions, restrictions or penalties as are prescribed by law and are necessary in a democratic society, in the interests of national security, territorial integrity or public safety, for the prevention of disorder or crime, for the protection of health or morals, for the protection of the reputation or rights of others, for preventing the disclosure of information received in confidence, or for maintaining the authority and impartiality of the judiciary.
Article 10 has never been used successfully against a private company or individual for preventing free speech. The general interpretation seems to be that it only covers public bodies (for example Wikipedia describes it as such and it's the stance taken by many anti-discrimination and non-profit groups).
It is generally used as an appeal against a lower court decision, so the case is held against the decision of the lower court, which represents the government. However the initial case does not have to be to do with government interfering with speech and can be to do with the actions of a company.
"In 2001 a journalist at the Financial Times (FT) received a copy of a leaked document about a possible company takeover by Interbrew. The FT published the document and three other newspapers (The Times, The Independent and The Guardian) and the news agency Reuters, also reported on the issue and referred to the leaked document. Interbrew brought proceedings against the news groups seeking to identify who leaked the document to the FT. The UK Court of Appeal ordered the FT’s journalistic source to be disclosed to Interbrew, using a common law principle (the Norwich Pharmacal principle) and the Contempt of Court Act.
The newspapers and news agency applied to the European Court of Human Rights, relying on the Article 10 right to freedom of expression. The Court held that the disclosure order interfered with the right to freedom of expression. The Court emphasised the chilling effect of journalists being seen to assist in the identification of anonymous sources. It found that Interbrew’s interests in finding the source of the leak did not outweigh the public interest in the protection of journalistic sources."
No, you would start with suing the company using the extensive freedom of expression protections in UK common law, then if the UK court didn't rule in your favour, but you felt it was in the wrong to do so, you then appeal against the court using the human rights act.
