Carroll clattered into De Gea on the stroke of half-time from a West Ham corner to leave Ferguson infuriated on the sidelines.
The striker, who is on loan from Liverpool escaped any punishment for the challenge on De Gea and Ferguson felt the referee got it wrong it letting Carroll go unpunished and that his players did not get enough protection from the officials throughout the game.
Asked for his opinion on the incident that sent De Gea tumbling to the floor, Ferguson told Sky Sports 2: "I think it's obvious, I don't think we need to dwell on that, it's an obvious red card but the referee has seen it differently.
"We know how they play, the ball is in the air most of the time. You've got to defend those things and they're very, very aggressive so you hope there's a strong referee. I'm not so sure we got that tonight."
Ferguson was happy with his side's fighting spirit as they battled back from two goals down to earn a point to move them a steo closer to the Premier League title and Ferguson felt it was a performance of "champions".
"We had to stand up to a lot, we went down twice but we kept going. We played like champions tonight," added Ferguson.
"It was magnificent in terms of determination and courage to play.
"It's one more game away...that's five left - three at home and two away. We're determined to win, we don't like losing and they showed that tonight again."