Another problem is Kakashi's failure to listen to his team.

Kakashi has no idea why Sasuke and Naruto were fighting on the rooftop to begin with. Kakashi guessed and assumed it was because Sasuke felt insecure about Naruto surpassing him in power levels. But Kakashi doesn't know that for a fact. Kakashi is still uninformed about precisely what conversation, dialogue, or incidents sparked the fight to begin with. Therefore Kakashi is running off of assumptions, not facts, and he hasn't bothered to ask any of the three Team 7 members their accounts, opinions, or feelings on the incident, which is a poor way to handle a problem of this type, because the only way to resolve a problem is to target its source.

Yet Kakashi hasn't bothered to collect information. He didn't ask Sakura, the neutral third party what happened, he didn't ask Naruto why he was fighting Sasuke when Sasuke was supposed to be getting bedrest in his hospital room, and he didn't ask Sasuke why he was fighting Naruto. Kakashi failed on all three sides, erroneously operating off assumptions instead of facts, which isn't a great place to start when handling social tensions and conflict, since it leads to incorrect judgments of thinking you have the perfect solution when in fact you have absolutely no clue what's going on. How can Kakashi treat the problem if he has no actual knowledge of how the incident started to begin with?

After all, ultimately the only one who can know what Sasuke is feeling is Sasuke. No one else can feel Sasuke's emotions but Sasuke. Outsiders can only guess and speculate, and while those guesses may be accurate or spot-on, it's no substitute for asking a person what they are feeling and trying to get the person experiencing those emotions to explain and define them.

So the best way for Kakashi to handle the situation would be to ask for Sasuke's explanation of events before jumping to conclusions.

That would provide Sasuke with a chance to defend or condemn himself of his own accord. Not only will this give a growing child the chance to independently assess his own actions and teach him to draw his own moral judgements on his actions as he is asked to overview his behavior/misbehavior (a tactic that may help a child learn introspection), it also gives them a sense of fairness in the judicial system around them, showing them that their input is valued, considered, and not entirely irrelevant or worthless.

Had Kakashi given Sasuke a chance to speak for himself and explain his perspective on what occurred during the incident, it gives a sense of fairness and genuine seeking of truth and justice on Kakashi's part. Instead of Kakashi making snap judgments about things without gathering the facts, Kakashi expressing interest in hearing Sasuke's version of events would signal that Kakashi isn't unfair or unjust, and he isn't seeking to make uninformed snap judgments, but also that he can't allow teammates to pettily fight each other, and infighting has to be punished firmly and severely.

Additionally, Kakashi also needed Naruto's and Sakura's versions of events.

Just as important to Sasuke' view of events is Naruto's and Sakura's translation of events. When two people fight, they are biased, often leading to irrational logic and biases marring an impartial and accurate version of events.

As the third party trying to stop the fight, Sakura is the most likely to have a neutral retelling of the events that occurred before and during Naruto and Sasuke's hospital skirmish.

All three genins' accounts were needed in order to fully piece together what the issue of conflict was and what punishment should be dished out, yet Kakashi didn't bother with any investigation into the matter, instead making the assumption that he knew what was going on.

This is a problem, because Kakashi doesn't know who instigated the fight and whether it was Sasuke or Naruto. For all Kakashi knows, Naruto could have started the fight (because typically Naruto tends to be the one starting and picking fights, not Sasuke). Kakashi demonstrates to his students that he doesn't care who started the fight; Sasuke takes the blame either way. If Team 7 misbehaves, Sasuke gets punished for it, regardless of who the actual culprit was.

Perhaps Kakashi made an educated guess in deducing that Sasuke is at fault by reading the room. And considering that Sasuke is in such a foul mood, Kakashi's educated guess may indeed just be common sense, especially since Sasuke doesn't deny Kakashi's accusations.

Furthermore, Kakashi needed to punish his team together. Naruto is at fault as much as Sasuke is and needs to be punished accordingly. It's only fair that either Kakashi punishes them at the same time, or at the very least relays to Sasuke that Naruto will likewise be punished and lectured accordingly.

By listening to each of the trio's account of events in turn, and punishing not just Sasuke but also Naruto, Kakashi would signal that he cares about ruling his team with fairness and that he values each team member's input. It would also signal that the team takes blame equally according to their actions, and Sasuke isn't the only one who gets punished when the team acts up.