The Argentine made headlines with his comments taking aim at the club’s hierarchy after the 2-0 defeat by Atletico Madrid.
The 28-year-old had criticised the board, namely director of football Jason Wilcox, for a lack of investment in the playing squad of the summer.
The defender then doubled down on his comments on social media before apologising on Thursday after a meeting with his manager and club secretary Omar Berrada.
"It's done,” Carrick said of the situation surrounding Martinez.
Making headlines: Lisandro Martinez
PA
“We've moved on from that. It was one of those things. As a group we've moved on. It's really not a drama for the group. The squad have trained really well and they are in good spirits.
"It was pretty simple. We understood the situation. Licha understood he made a mistake. We have a really good felling from the club. Results wise it hasn't gone our way, but the spirit and togetherness. We can see in the group we want to do something with this situation.
"We've moved on. Genuinely. There is a freshness about the group.
"He [Martinez] is available."
Reports in recent days have suggested that Martinez’s future at Old Trafford is in doubt after his comments.
Furthermore, it is believed that the Argentina international pushed to leave the club in the summer, wanting to move to Atletico Madrid amid displeasure over not being offered a new contract.
The defender’s current deal in the north-west expires at the end of this season.