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United were trailing to Noah Okafor’s first-half double when Martinez was dismissed.
And although Casemiro got one back for Carrick’s side, they were unable to avoid the second defeat of his 11-match reign.
“I thought the boys, the way they went about it, they stayed positive and fought to get something out of it after another shocking decision to send off Lisandro,” Carrick said.
“Two games in a row we’ve had decisions like that go against us but this one was one of the worst I’ve seen.”
Claiming Calvert-Lewin’s initial contact with Martinez had played a role in the centre-back’s reaction, Carrick said: “You can throw your arm in Martinez’s face and then as he’s off balance because of that, he’s half grappling, he half touches the back of his hair which pulls the bobble to come out.
“I don’t even know what it looks like. It’s not a pull, it’s not a tug, it’s not aggressive. He touches it and he gets sent off.
“Worst of all, he gets sent to overturn it, a clear and obvious error. Shocking.”
Martinez could face a three-match ban and, asked if United would appeal, Carrick said: “I am going to have to discuss it but I think it is a bad decision.”
Carrick was also frustrated that Calvert-Lewin got away with a strong challenge on United defender Leny Yoro in the build-up to Okafor’s opener.
“We obviously conceded when Leny Yoro gets a forearm smash in the back of the head and they score the first goal,” he said.
“They didn’t decide to overturn that decision. That was a big moment in the game.”
United remain in third place in the Premier League, seven points ahead of sixth-placed Chelsea in the race to qualify for the Champions League via a top-five finish.
With one win in their last four games, United travel to Chelsea on Saturday for a crucial clash in the battle for Champions League places.
“This is one game. We’ll certainly look at it and look to improve. We’ve got another big game, an important game next week and a big end to the season,” Carrick said.
“We’re in a really good position. Tonight’s disappointing, we didn’t want to lose and we didn’t want the referee’s decisions to be so bad against us but we move on.
“We’ve got a big end to the season and there’s a lot to be positive about.”