Solskjaer was sacked on Sunday
United parted company with their 1999 treble hero in the wake of their 4-1 defeat at Watford and have put Michael Carrick in temporary charge while they seek to appoint an interim manager to the end of the season.
The decision comes at the end of a torrid run of results for United under Solskjaer, who himself took over on an interim basis back in December 2018 after the sacking of Jose Mourinho.
He rode out a chastening 5-0 home loss to Liverpool and appeared to be back in calmer waters after a 3-0 win at Tottenham, but a comprehensive derby loss to Manchester City meant that the Watford game was essentially a must-win.
Derby manager Rooney said: “I was so disappointed watching that game yesterday because, as a manager, you can set a team up and work on shape and how you want the team to play, but there’s no excuse for some of those performances from some players.
“To see players waving their arms around and giving the ball away but blaming other people was not acceptable. I would be very angry if I saw that from my players. Those players are at one of, if not the biggest, club in the world and should feel privileged to be there.
“In my opinion, the players have to show more – that doesn’t necessarily mean winning the Premier League or the Champions League, but showing the fans that they are working hard for the club.”
Asked about being linked with a coaching role at Old Trafford, Rooney added: “I’m committed to this football club until somebody above me tells me otherwise.
“I speak to people at Manchester United on a regular basis and I’m sure, if they were going to come calling for me, that would have happened by now.”
Gary Neville praised former team-mate Solskjaer for restoring the “soul” of the club, but admitted the Norwegian’s exit had been on the cards.
“It’s been coming for the past few weeks, performance terrible and results shocking. Yesterday was wimpish at Watford,” Neville said on Sky Sports.
“You’re always hoping the players would respond, that the two-week international break would bring some freshness but it looked like they had the world on their shoulders, their performance levels over the last few weeks have dipped.
“When a manager can’t get a performance out of his players and results are getting as bad as they are, in this game you’re going to lose your job.
“The worst thing is the performances, the team have looked all over the place. Yesterday, the highlights, some of the goals, the defending is absolutely woeful and that’s a back four, five that have played together a number of times.
“Watford aren’t the best team in the league by a long stretch but they got mauled, Manchester United. Ole couldn’t get a performance out of them in the end, the players looked drained of confidence and I’m not surprised it’s ended.
“A couple of weeks ago I thought getting to the end of the season was the right thing to do if they could, they obviously haven’t been able to do that.”
Former United defender Rio Ferdinand told BT Sport: “This has been inevitable, the way the team has kind of gone backwards since the start of the season after a very excitable transfer window with some big names coming in.
“It is not the way we wanted it to end, the way we wanted it to go. It has looked inevitable the last few weeks especially the way Man Utd lost at home to Liverpool in the manner they did and again in the manner they did to Man City. I thought maybe then would be the time (for Solskjaer to leave).
“Disappointed obviously in the way the team has performed. It is unfortunate for someone like Ole but it is the way football goes, it is a cut-throat industry and if thing aren’t going according to plan, big decisions are made. We have seen one made this morning.
Peter Schmeichel, who won the treble alongside Solskjaer in 1999, said he did a “great” job in difficult circumstances but says United have to get their next appointment right.
“It is a very sad day,” the Danish goalkeeper told Sky Sports News. “It’s the fourth time we have to sack a manager (since Alex Ferguson retired) so clearly something is not going right at the club.
“Hopefully, the fifth time we get the right person in to get the club organised in the right way.”