 Speed has been irked by May's strong comments |
Cricket's governing body has become embroiled in a row with the umbrella organisation for players' unions. The Federation of International Cricketers Associations has criticised the International Cricket Council over the amount of cricket being played.
But the ICC has hit back by accusing Fica of a "lack of respect" and adopting a "belligerent" approach.
It insists such matters should only involve individual cricket boards and players' unions in those countries.
The row began after cricket's bosses agreed to ratify the future tours programme, which the ICC said had been modified to address concerns about player burnout.
The cycle of Test teams playing each other home and away has been extended from five to six years, while the lengths of the World Cup and Champions Trophy tournaments have been reduced.
 | More than ever we are seeing top players forced to retire as a result of the unrelenting schedule |
But May says the decision to introduce a Twenty20 World Championship, the first edition of which is likely to take place in England in 2009, shows they have not heeded the concerns.
"Instead of easing player workloads, the programme commits more matches to be played," the former Australia spinner claimed.
"It will result in more cricket for a number of teams over the next six years than we have ever seen before.
"More than ever we are seeing top players forced to retire from the demands of the game as a result of the constant and unrelenting schedule.
"Injuries and forced absences to fast bowlers are becoming more pronounced, and some teams are forced to apply rotational systems to ensure players are not flattened by these congested playing schedules."
However, the ICC responded by questioning the legitimacy of Fica in such debates.
Speed said: "The ICC board is concerned that Fica is currently representative of only five player associations which are recognised by their respective boards.
"Concerns have also been raised that Fica is attempting to become involved with issues which are the domain of individual boards and their player associations."