Assistant manager Carver was caught on camera shouting at supporters displaying a banner calling for manager Alan Pardew to be sacked prior to the 4-0 defeat at St Mary's.
Woodman allegedly then took to Twitter to abuse a disgruntled fan, with a statement on the club's website appearing to confirm the suggestions.
Carver said: "My sole focus on Saturday, as it always is, was on trying to gain three points for Newcastle United and preparing the players to the best of my ability.
"This included keeping them focussed on their roles and duties in the upcoming 90 minutes.
"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."
Woodman also issued an apology, 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."