I suspect all the weebs know but figured I'd mention anyway:
In Japan, it is common when someone is injured for the parties to negotiate monetary compensation; its possible for courts to get involved but this considered shameful for everyone involved (this is one of the reasons for the low number of lawyers in Japan
this is also why there is such low reported rape/sexual assault as most cases are handled non-judicially & quietly with what's effectively an admission of guilt and payment of hushmoney but lets not go into that). Anything "permanent" is deemed pretty serious. Hence "Taking responsibility (for the injury)" is a serious thing.
As we also know, the term also relates to marriages, especially of the Shotgun kinda, as if a woman is "ruined" it is expected the family that did the ruining takes her in; you break it you bought it rules.
So the nurse is likely teasing with a double entendre, Homura (being a poor teacher) is making his own unintentionally and poor Hasumi is left with her heart a-flutter.
He was pretty much holding her hand while helping her clean up the blood. So they're basically married now right? Or it's at the very least it's some sort of blood pact.
Married and they are likely both pregnant with that much unprotected handholding