St Mirren accepted the immediate suspension offered by Scottish Football Association compliance officer Vincent Lunny, who cited Goodwin for violent conduct "by striking an opposing player, namely Stuart Armstrong of Dundee United FC, on the face with an arm".
But they hit out at their Scottish Premiership rivals, who posted stand-alone footage of the incident on their YouTube page with multiple replays and links to their Twitter and Facebook accounts.
Dundee United manager Jackie McNamara had earlier claimed Goodwin could have been sent off before he was substituted in the 43rd minute of his side's 4-1 victory.
A statement from St Mirren read: "As a club we were disappointed to find one of our players singled out for criticism by another football club both in the media and thereafter on their website. We accept that football is a game of opinions but we find this unacceptable.
"As a club, we have on many occasions been aggrieved by decisions, tackles and challenges made on our players. We have, however, dealt with such matters behind closed doors, via the appropriate channels and in a professional manner.
"We are of the view that one club should not be commenting on another club's players in public and feel that these issues should be dealt with by the governing body (Scottish FA) to protect the integrity of the game."
Before the ban was offered, Saints boss Danny Lennon admitted his surprise at United's actions.
Lennon said: "My football club would not do that. It's not in the make-up of our club to be putting anything out there (on social media).
"With the very good coverage that the BBC give our game, nothing would go unseen by the SFA and the disciplinary panel.
"So I found it a little bit strange that another club would do that. We all work in different ways but that was a first for me."
Goodwin was back in the Saints side that drew 1-1 with Hearts at home on Sunday but will now miss league games against Kilmarnock and Celtic this week.
Lennon said: "Jim has fantastic qualities but like anyone he has weaknesses. However, what he brings to this football club as captain and by being a terrific guy off the pitch and a winner on it is great, so I'd rather focus on his qualities."