There is no legal requirement for a secret ballot unless rejection of a pay offer is tied in with an intention to take industrial action.
If the vote was taken solely among members of a recognised trade union, then the union's own rules may state that a secret ballot should have been organised. Otherwise it would have been up to the employees involved in the voting procedure to determine their own method of voting. (e.g. they could set up a 'works council' to deal with pay negotiations. The voting rules would then be determined by the constitution and procedures of that council).
It would probably be impossible to draft a law stating that all pay offers should be put to a secret ballot. Many smaller firms don't have formal negotiating machinery and, therefore, never make any pay offers - they just pay what they can afford. Even where a company does make a formal pay offer to its employees there might be members of several different unions involved (+ non-union members). There could be all sorts of difficulties working out whether it union should conduct its own vote (in which case, who represents the non-union members?) or whether there should be a vote for the whole work-force (in which case who should organise the voting?).
Chris