The Red Devils suffered a disappointing 2-1 defeat to the Cottagers in the Premier League last weekend, with Alex Iwobi scoring a 97th-minute winner at Old Trafford.
After the match, Fulham uploaded a video on TikTok of Fernandes falling to the ground and clutching his leg following a challenge from Sasa Lukic.
As play continued without a free kick being awarded, the Red Devils captain swiftly got back up onto his feet to make himself available for a pass on the edge of the penalty box.
The 15-second clip is accompanied with comical music and the caption "So glad he's OK" as well as the eye-rolling emoji.
Ten Hag was made aware of Fulham's post during a press conference on Thursday, and although at the time he had not seen the clip, he has encouraged the Cottagers to apologise for the "totally out of order" video.
Asked directly about Fulham's video, Ten Hag told reporters: "I didn't know this, but if they did it I would say it is not right.
"It is absolutely not right that a club makes such a statement, because it is totally out of order and they were wrong so they should apologise for this."
Fernandes emerged as a doubt for Man United's FA Cup fifth-round tie against Nottingham Forest on Wednesday, and according to The Telegraph, Man United sources claim that the incident in which Lukic caught the 29-year-old was the moment he sustained his injury.
Man United's No.8 was eventually fit to start the contest at the City Ground and his free kick set up Casemiro to score an 89th-minute winner in a 1-0 victory, setting up a quarter-final clash with Liverpool.
However, Fernandes was seen limping out of the City Ground after the match and has now emerged as an injury doubt for Sunday's Manchester derby at the Etihad Stadium.
Fernandes had his throat grabbed by Forest centre-back Felipe towards the end of Wednesday's match, in which he was also on the receiving end of several strong challenges.
Asked if Fernandes needs more protection from referees, Ten Hag said: "He is a very passionate player. He is a very creative player. He is creating the most chances in the Premier League.
"You see opponents targeting him, especially after Saturday when he had the knock and they see it. Then I feel the refereeing should protect him."
Fernandes has played the most minutes (3,085) of any outfield player at Man United this season across all competitions, scoring seven goals and registering eight assists in 35 appearances.