This!!
"It sounds like she's making it up"
Well BIL's a little Iago, isn't he? Whispering adverse comments in everyone's ear as long as the target never finds out.
Stand up for yourself otherwise, there will be lingering resentment and it will fester, you will always be second-guessing yourself and wondering if they are all criticising you behind your back, making you underconfident and unsure of yourself in family situations. It could damage your future relationships with DH family and impact your relationship with DH if you let it fester.
He's actually telling all of them that you would lie about DD's serious health condition for the drama and attention. It's tantamount to saying to them, "This is what she's like so don't believe her future appeals for help or sympathy."
It has an influence on how the rest of them view you going forward, how they will respond to things you tell them, so you must challenge it. He's nasty.
You are a couple, a unit, and this attitude and adverse PR affects both of you. I think DH WOULD want to know. It's his DD too.
I wouldn't bother asking BIL or anyone else for an insincere apology.
But tell him straight out.
"You are very happy to ask for our help when you need it, so please Stop spreading rumors about me behind my back, if you've got something to say, be an adult and say it directly to my face, for the sake of accuracy. You'll be happy to know that after her hospital stay, DD is recovering." That is not falling out with him, but standing up for yourself and calling out his shitty behaviour.
I'd also be asking
- DH who else is on this group? Is DH on it. How long have these comments been going on? DH? What does he think about this. Is he going to back your request, that they make any accusations about you to your face. You don't have to have him as your spokesperson/intermediary, you can speak for yourself but it would be nice if he backs you up.
- I'd take it up with MIL too. She'd seen these messages but said nothing. Did she defend you? I think there's a 60 percent chance she realised you'd seen it.
You don't have to be aggressive. There's no need to be a family fall out. Do it calmly, and Be very matter-of-fact, leaving out recrimination and accusations.
Tell MIL that you know , what does she think about it? They are all welcome to their opinions, but if they have a problem with your behaviour, have the goodness to say it directly to you and give you an opportunity to respond, rather than spread inaccurate stories behind your back. I think it's better to say this rather than let these feelings fester and damage future events.
Also, your the unofficial family drama queen according to BIL, so keep all personal/medical posts off Facebook from now on (its a good idea anyway and- that's what messenger is for) And I'd quietly block BIL on Facebook and other social media from now on if anyone asks - you thought he didn't like seeing your "drama". Also. Be confident in yourself. Who cares what they think? So stop worrying about them.
Sorry OP I know that this would be hard, and its easier said than done.