If you're wondering what narrative purpose this scene serves, you have to consider the whole series. It's a mirror of a scene in episode one where Shinji also stands on top of a female character with a bodily fluid on his hand, but for a whole other reason: Shinji was told to pilote a giant robot he'd never seen before to fight a giant monster. He refused. The injured girl was brought on a strainer and he was told if he didn't pilote the robot, she'd have to. The ground shook because of the robot, the girl (named Rei) fell from the strainer and Shinji rushed to see if she was okay. He looked at his hand and saw that he had her blood on it (obvious symbolism), then he accepts to pilote the robot.

That scene is what asserts Shinji as a protagonist. It's the first showing him doing something for someone else, and he's putting his life on the line to do so.

So mirroring this scene but having him do something cowardly and shameful, opposite of the bravery and kindness he showed in episode 1 makes him exit this role as a protagonist. And I don't even think it's necessary to understand it for it to work: For most of the movie after that, Shinji isn't the protagonist, you follow other characters as they conclude their respective narrative arcs and without the hospital scene, I think many would feel frustrated and wait for Shinji to do something. Instead, we're more prone to watch the other characters because we don't really want to see Shinji anymore.

Another thing is, there characters have all been through a lot and been repeatedly traumatised. Not that it's an excuse, but the series is also a bit original in the way it rejects the trope that hardship builds character and makes one better, without going in the reverse cliche of it making them a villain either. Trauma makes them mentally ill. Mental illness sometimes cause them to do bad things to other, which they then regret.

The whole movie, even the whole series was pretty fucked up like this.