Wigan forward McManaman escaped punishment during Sunday's game despite a high challenge which forced Haidara off with a knee injury.
Referee Mark Halsey's report confirmed one or more of the officials had seen the incident and an FA statement said its current rules - agreed with all the game's "stakeholders" before the season began - allow for no retrospective action.
That has angered Newcastle, whose managing director Derek Llambias accused the FA of lacking "the courtesy to contact the club and let us know."
A statement from Llambias read: "It is clear from this decision that the current disciplinary procedures are not fit for purpose.
"Newcastle United, along with other clubs, have had players suspended for incidents reviewed after the game. Whilst not trivialising these incidents, they were not, in our opinion, of the seriousness of Callum McManaman's tackle on Haidara.
"Whilst we understand that the current procedures give the FA limited options, it cannot be correct that the most serious offences can go unpunished, even if the original incident was seen by match officials.
"We will now be making a strong representation to the FA and the Premier League to see how a more appropriate, fair and even-handed disciplinary process can be introduced at the earliest opportunity."
Llambias was also critical of Wigan owner Dave Whelan, who called McManaman's tackle "fair" and claimed the player had won the ball "as clean as a whistle".
"It was not a fair challenge," he said "This view is shared by countless former players, referees and well-respected media commentators. Indeed it appears to be only Dave Whelan who takes a contrary view."
Haidara, meanwhile, is waiting to discover the full extent of his injury, and Llambias said: "Contrary to comments in the media, there is currently no timescale for his recovery."
No retrospective action
A statement from the Football Association said that "because at least one of the match officials saw the coming together, though not the full extent of the challenge" no further action can be taken.
The statement read: "Following consultation with the game's stakeholders in the summer, it was agreed that retrospective action should only be taken in respect of incidents which have not been seen by the match officials.
"Where one of the officials has seen a coming together of players, no retrospective action should be taken, regardless of whether he or she witnessed the full or particular nature of the challenge. This is to avoid the re-refereeing of incidents.
"In the case of McManaman, it has been confirmed that at least one of the match officials saw the coming together, though not the full extent of the challenge. In these circumstances retrospective action cannot be taken.
"The principal objective behind the not seen policy is to address off the ball incidents where match officials are unlikely to be in a position to witness misconduct."