The FA confirmed that the incident was seen by at least one of the match officials - even though referee Mark Halsey was unsighted - and under current rules that means retrospective action cannot be taken.
The announcement led to a storm of criticism from former players and pundits, and Newcastle's managing director Derek Llambias said the club would make immediate moves to change the disciplinary process.
To further infuriate the Magpies, assistant manager John Carver has been charged with misconduct by the FA, along with Wigan coach Graham Barrow, following a clash at half-time in Sunday's match.
Llambias said the disciplinary process was "not fit for purpose".
Haidara suffered what could prove to be a serious knee injury after the challenge by McManaman, but no action was taken against the Wigan player during the match.
Llambias said in a statement: "We are disappointed to learn that the FA is not going to charge the Wigan player.
"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 - those which have the potential to cause another player serious harm - 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 to prevent incidents of this nature going unpunished in the future."
The FA's rules follow a decision taken last summer - after consulting with the leagues, Professional Footballers' Association and referees - limiting retrospective action to off-the-ball incidents unseen by match officials. The rules apply unless there are exceptional circumstances, which would need to be more unusual than a late tackle, however nasty.
The FA said in a statement: "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 Callum 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."
Llambias also responded to claims by Wigan chairman Dave Whelan that it was a fair challenge, saying the comments had "disappointed and surprised" him as they came from someone he respected.
Llambias added: "It is our strongly held opinion that the tackle on Massadio was extremely dangerous and is the type of challenge that has the potential to cause serious harm and such was the force, and reckless and dangerous nature of the challenge, even end a player's career.
"It was not a fair challenge. 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."
Llambias said Haidara had a scan yesterday and will have another one next week to assess the extent of the damage.
Joey Barton, who was given a 12-match ban after clashing with Manchester City players after being sent off for QPR last season, claimed the FA were not consistent.
The QPR player, currently on loan to Marseille, wrote on Twitter: "I am almost certain if I or another player with a negative FA persona had made that challenge, I'd [have] without doubt received a ban."