The 37-year-old footballer joined the Red Devil from Chelsea in 2017 to reunite with former Blues manager Jose Mourinho, but he threatened to leave the club after a disagreement when bosses tried to punish him over the incident.
He told FourFourTwo magazine: "On the Wednesday, I didn't train because my ankle was swollen and I said to the physio, 'Can you do the treatment this afternoon?'
"We did that, but there was a commercial day at Old Trafford at 3pm.
"A couple of days later, my agent called and asked what I'd done, as I didn't check my emails. I had a look at I'd been fined for not going to the event.
"I went to Jose and said, 'Listen, if I pay this, I'll never club for this club again.' He said he'd get it sorted and it was, but something like that would never have happened at Chelsea.
"I don't want to criticise United, I respect them deeply. I hope it's different now.' "
Matic suggested United's hierarchy were too "focused" on commercial performance, while he insisted "a lot of the players" played to a high standard during his five-year spell.
He said: "I was going to United a champion and was motivated to be the same there.
"When I saw the team, I was sure we'd fight for the title, but Manchester City became champions and we finished second.
"I played well - a lot of players did. I believed we could fight properly for the title in my second season, but I do think the club, was focused on the commercial side."
While linking up with Jose again was a big selling point for the move, Matic revealed he also held United as a whole in high regard, which helped influence his decision to sign.
He explained: "United is a big club, a fantastic club, and to be honest, I felt like I needed to have United on my CV.
"I felt very honoured and privileged to represent them and, even today, I still think I made the right decision.
"For me, United and Real Madrid are historically the two biggest clubs."