The argument about whether the press can publish about relationships and gossip is not an argument about public interest; it is clear that the public don't really have a need or right to know who is shagging who, at least not directly.
However, it can be argued that the injunction conflicts with Freedom of Expression. Essentially the injunction brings into play where the balance between a right to Privacy and Freedom of Expression lie.
The European Convention on Human Rights gives states a wide "margin of appreciation" on where they may place the dividing line. For example, should the US ever become a signatory, it could legitimately claim compliance with the convention even though Freedom of Speech is regarded as a more important right than Privacy. In France, on the other hand, more importance is given to privacy. As long as both rights are protected to some extent, both nations would be regarded as compliant in terms of their implementation.
Censoring the Web is fraught with difficulty and could be challenged under those same rights.
When the UK implemented its Human Rights Act, which effectively ratified the ECHR in the UK, it was openly stated by Parliament that a high threshold would be required for matters of Privacy to overcome matters of Free Speech, i.e. Parliament wanted a Human Rights implementation closer to the US than France. Unfortunately, legal rulings have been quietly lowering the bar in clear contravention of this intention.