United appointed Carrick on an interim basis following Ruben Amorim's sacking in January before handing him a contract until 2028 after he oversaw a turnaround in the side's fortunes.
However, United's decision to sack Carrick after one league win this season, leaving them 18th on eight points after 10 games, comes amid supporter anger over the team's problems.
The Glazer family, which still owns a majority of the club, has faced strong criticism from fans for overspending on average players and putting off investments in infrastructure.
"It is time for the owners to explain themselves and show some accountability. But the discontent at United is not just about matters on the pitch. Far from it," the trust said in a statement.
"Aggressive ticketing policies have further damaged relations with supporters... Supporters are being asked to pay the price for the owners’ mismanagement as well as their stadium ambitions through higher charges and diminished rights."
In July, United unveiled the proposed location of their new, 100,000-capacity stadium, next to the existing Old Trafford. According to the club, the flagship project of minority owner Jim Ratcliffe could create 15,000 homes and contribute more than £7 billion a year to the British economy.
"The owners must allow senior club management to change direction: show accountability, rebuild trust, listen to supporters and recognise that their loyalty is fundamental to football success. Another change of head coach cannot repair what the ownership continues to undermine," the trust added.
Reuters has contacted United for comment.
Carrick is the seventh permanent manager United's hierarchy has sacked since Alex Ferguson clinched the club's 20th league title and retired in 2013. He is also the third manager sacked since Ratcliffe came on board in 2024.
USA national team manager Mauricio Pochettino is likely to take over from Carrick, British media have reported.
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