I have a bit of an insider's view on this, having done CPS for 8 years previous to my current gig as a hospital social worker on an LDRP. I'm not in CA, though, and each state has its own code and policies outlining child protection mandates...
In the hospital where I work, it's pretty common that a mom and baby can be discharged after 24hrs after an uneventful vag delivery, pending no other subsequent baby issues (ex, GBS +, phototherapy needs, etc). The OB clears mom for discharge, and the ped clears baby. In our hospital, this ped can be your community-based ped (if he/she has hospital privileges) or one of the hospitalists peds (if your ped does not have privileges). While some OBs will discharge a mama <24hrs, most, if not all, peds will want to wait for the 24hr mark.
There have been instances in which a mom wishes to discharge with her baby <24hrs, and the ped usually consults me to chat with the parents about their desire to leave in this "quick" timeframe (for hospital-based births). It's true, I am mandated by hospital policy to notify the dept of social services where the family lives. Typically this means I speak with a CPS social worker and explain the context of the situation. If there are no other red flags (like a history of abuse with the family or any previous CPS encounters, or concerning behaviors exhibited by the family during their stat, such as being inattentive to baby's needs), usually CPS will want to know who the baby's ped is and when the baby's follow-up is scheduled. Sometimes they will touch base with the family to explore the situation further, but rarely have I seen a circumstance where CPS advises the hospital that a baby cannot go home with parents or must stay. In those circumstances where CPS has formally intervened and prevented a discharge home, there is usually some outlying issue with the family.
I have found that it really is all in how you approach this subject with the family. If the ped or hospital social worker barge into a room and treat a family like they're a bunch of inconsiderate idiots, well, that's just asking for an outraged family. But if you gently explain that this is a mechanism that the hospital must engage to ensure the safety of babies and is in no an indictment of a parent's intent or ability to be a nurturing parent, most parents handle this little hiccup pretty well. I agree with an above poster that it's important to realize that a hospital has an obligation to ensure that a baby is safely discharged to the parents, and there are certain procedures that the hospital must adhere to legally...because there are, sadly, families that are not as thoughtful or as prepared as yours. I've seen some yucky stuff, and it's a shame that other families get lumped in with that.
Having said that, my advice to you is to be a proactive as you possibly can. 1)Touch base with your local dept of social services and discuss your intentions as your EDD rolls nearer. Chances are, if you have already placed this concern on their radar, they may be less likely to be super reactive when/if the hospital touches base with them. 2)Touch base with the hospital, possibly the nurse manager on the postpartum unit and explore this further. Let her/him know you've been in touch with DSS about this concern. Let them see that you're a thoughtful and prepared parent. Inquire if there is anything they might be able to do to proactively smooth out any wrinkles... 3)Find out who is going to be the ped responsible for discharging your baby--a hospital-based ped or your community ped--and discuss this further. Possibly even have this ped draw up a letter that can be placed in your prenatal records that go with you to the LDRP.
The worst possible thing you can do in this circumstance is nothing and waiting until you've delivered to explore the logistics of your early discharge. Hospitals, as you know, run like a factory (sad, but true) and anything out of the ordinary sends folks into a panic, and sometimes a very judgey panic at that. And you don't need that stress on such a big milestone day!