MacAnthony on Wednesday hit out at Wright and the Perth club over their handling of the transfer saga.
St Johnstone rejected the terms of Posh's bid but May has now ruled out a move to the Sky Bet League One club himself.
Responding to MacAnthony's comments, Wright said: "We tend to do our talking through proper media channels. We don't need to be going into social media, I don't think that's the right way to do it.
"Some of the things he says could be quite easily contradicted but he can rant and say all he wants, Stevie May won't be going to Peterborough.
"I had a short conversation with Stevie yesterday and he told me he wouldn't be going to Peterborough so maybe that's why their rant came about.
"But I have a football club to run and he's got a Twitter account to run so we'll just leave it at that."
Writing on his Twitter account on Wednesday, MacAnthony said: "Jeez like how many press articles has the St Johnstone manager done about S May & us.
"Enough already as its (sic) getting silly. So some facts: As revealed by him, yes we put bids in for said player.
"They rejected them, sent us their terms, we met all terms bar how they want money paid.
"They want it all in this budget year. We never do our business that way as we have told them & have % rules etc to work within.
"Posh will not be changing final offer made last week & no amount of press articles will make a difference.
"And as for the 'No disrespect, he might get a bigger club looking at him' comment, I hope that works out for them & their club."
May has scored 51 goals in 92 first-team appearances, most of them during loan spells with Alloa and Hamilton. The forward signed a one-year contract this summer and has netted three times this season.
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