The 49-year-old approached the Magpies' travelling fans at St Mary's, reacting angrily to the sight of anti-Alan Pardew banners in the crowd.
Following a "number of complaints" made to the club, Newcastle say that they have now dealt with the matter internally.
In addition to a short statement released by the St James' Park outfit, Carver said: "In the warm-up, I was disappointed to see banners that could, in my opinion, distract the players and take focus away from the job in hand - namely winning the match.
"I went over to our fans to ask them to help us and requested that they take the banners down. The vast majority of supporters back the team through thick and thin and I am very proud to be one of them. I was met with abuse from one supporter and I regret that I responded inappropriately. I sincerely apologise for any offence caused by my reaction.
"Every fan is entitled to an opinion, positive or negative, and I respect that. I hope fans will understand that I, like them, only wanted what was best for the team. I am proud to work for my home town club and would ask every supporter to back the players and all staff, as they really can make a difference."
Goalkeeping coach Andy Woodman also issued an apology for his part in the events, adding: "I am very sorry for the ill-advised comment I posted privately on Twitter on Sunday morning, which was subsequently made public.
"It was never my intention, of course, for my response to enter the public domain. My response was a reaction to significant personal abuse I received from a number of fans which I felt exceeded that which was fair and balanced.
"That said, I understand my response was unacceptable and I regret my actions and apologise for any offence caused."
Newcastle are back in Premier League action this weekend when they take on Hull City.