Goodwin was handed a notice of complaint by Scottish Football Association compliance officer Vincent Lunny over his challenge on the United midfielder during his side's 4-1 victory on Boxing Day.
The midfielder, whose raised arm felled Armstrong as he won the ball with a tackle, has until 3pm on Tuesday to respond to the charge and will miss Scottish Premiership fixtures with Kilmarnock and Celtic unless he challenges the ban.
But Lennon is angry that the Tannadice club posted stand-alone footage of the incident on their YouTube page, with links on their Twitter and Facebook accounts, in an apparent bid to draw the attention of Lunny.
And Lennon insists it is a ploy St Mirren would never consider.
The Buddies boss told Press Association Sport: "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."
Lennon had to substitute his skipper in the first half before he was sent off against United but Goodwin was back in the Saints side that drew 1-1 with Hearts at home on Sunday.
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."
United manager Jackie McNamara felt Goodwin could have been sent off before he was substituted.
In his post-match interview, McNamara said: "I was disappointed with some of the challenges that went in.
"That was reflected in the substitutions that Danny made before the end of the half. Jim could have been sent off before that.
"It wasn't just the number of fouls but the manner in which some of them were carried out."