Rooney, United's all-time top scorer, spoke after United finished sixth in the Premier League with 66 points, the fourth time in six seasons they've failed to reach 70.
He said on The Wayne Rooney Podcast (h/t MailOnline's Adrian Kajumba): "I think the players need to fear someone. They need to fear Ole Gunnar. They need to fear Michael Carrick. They need to respect them but fear them also."
The DC United forward said the players need to take more responsibility, and he also said he finds it "remarkable" they advertise their brands on social media following defeats:
"When fans speak up on it and say 'why are you posting that?' they always have the marketing people to blame [but] take responsibility.
"They work for you. Those marketing people work for you. I have people who do similar stuff for me and they never do anything without my instructions.
"If that is what you're doing, you're sitting at the top of that business. You have to take responsibility for them.
"These players almost always like to find someone to hide behind, whether that is on their social media or on the football pitch and that's what they are doing."
Solskjaer replaced Jose Mourinho in December after United took just 26 points from 17 Premier League games under the Portuguese manager.
The Norwegian was a breath of fresh air compared to Mourinho's man-management, tactical approach and dealings with the media, and the players evidently responded as they won 14 of his first 17 matches in charge in all competitions.
However, they were unable to maintain that return to form and reverted back to struggling.
Football writer Liam Canning put their 2-0 defeat to Cardiff City on the final day of the Premier League season in perspective:
Bleacher Report's Rob Blanchette wants to see change at Old Trafford ahead of next season, including Solskjaer's management and leadership of the team:
The 46-year-old's positive approach yielded impressive short-term results, but for United to improve long-term, he needs to find the right balance of relating with his players and maintaining authority over them.
As Rooney noted, the players also need to take responsibility for their own performances, too, as few—if any—of them played to their potential this season.
Mourinho got 81 points in his last full campaign in charge with largely the same personnel.
Given Manchester City and Liverpool got 98 and 97 points respectively this season, that tally likely won't be enough to mount a sustained title challenge next season, but it would at least put them back on an upward trajectory and give them an excellent chance of a top-four finish.
United will need their summer recruits to hit the ground running for that to happen, but they'll also require improved showings from the players who remain at Old Trafford over the summer.