It's also worth noting that the child will likely need a social security number, which requires either implicit (applied for at hospital of birth) or explicit proof of citizenship.

@phoog As a matter of fact, it may not even depend on the law in mom's home country. Japanese law requires dual citizens to "choose or lose" by age 22 -- if a dual citizens does not renounce her other citizenship by age 22, she automatically loses her Japanese citizenship. Yet as far as anyone can tell, this clause has never been applied!

@Eric as far as I understand it, the child does not need a social security number, until he or she starts to work and pay taxes, if anon does not claim an income tax exemption for the child. Depending on the value anon places on keeping the child out of the US government's sights, it might be thought worthwhile not to claim the child as a dependent. (It's not the decision I would make, to be clear.)

@phoog Signing the child up for health insurance is likely to invite requests for his/her SSN. Agree that this can probably be avoided but the day-to-day headaches of not having a SSN will likely be felt sooner than any decision for or against dual citizenship.

@Eric I have no recent experience with health insurance for dependents without social security numbers, but it was entirely unexceptional 30 years ago. Even today, it must be possible because people can have foreign family members without SSNs who must be insurable.

the OP said they plan to move to EU after a few years. Wouldn't the child require some kind of proof of citizenship for his travel documents at that time anyway?

@user17915 yes, but an EU passport would be sufficient in practice ("in practice" because leaving the US without a US passport would violate 8 USC 1185(b), but doing so has no penalty, airlines do not check, and CBP officers almost never inspect departing passengers), so it would be possible to have only proof of the EU citizenship.

I know its just semantics, but the OP can choose not to register the child. It just doesn't make any difference as far as the United States' recognition of the child as citizen. There is no legal requirement to register the child and the parent cannot be compelled to do so.

@ouflak And, depending on what you mean by "register the child," there may be no way to do so! There is no registry of citizens in the US. There is no national registry of births. However, I believe all states legally require parents to file a birth certificate. Who they file it with, and what is done with that information, varies across states.

@Justaguy, "...However, I believe all states legally require parents to file a birth certificate..." In fact, the opposite is true! Birth certificates are given at the discretion of the registering authority (state usually, but can be a county or other jurisdiction). Hospitals (and I suppose midwives, paramedics, etc...) are indirectly under some obligations since they have to record all medical treatments under a variety of reporting laws. Other than that, a parent never has to request a birth certificate, much less 'file' one.

@Justaguy, Clearly that fact doesn't take into account such later practicalities as attending a public school, getting a passport, insurance, playing sports, and so on....

@outflak "a parent never" Interesting. In the US, parent = father or mother, as in, RCW 70.58.080(6) If there is no attending physician or midwife, the father or mother of the child, householder or owner of the premises, manager or superintendent of the public or private institution in which the birth occurred, shall notify the local registrar, within ten days after the birth, of the fact of the birth, and the local registrar shall secure the necessary information and signature to make a proper certificate of birth.

@outflak "Hospitals....are indirectly under some obligations" In at least one state, the obligation is direct: RCW 70.58.080(1) Within ten days after the birth of any child, the attending physician, midwife, or his or her agent shall: (a)(a) Fill out a certificate of birth..(b) File the certificate of birth together with the mother's and father's social security numbers with the state registrar of vital statistics."

Of course, if OP really really doesn't want their kid to be a US citizen, they can renounce their own US citizenship and have the kid born outside the US.

@phoog Is 8 USC 1401(e) a typo? The link goes to (g), and (e) covers people born in "outlying possessions." I'm pretty sure that doesn't include the EU, at least not yet. So I'm pretty sure you meant 8 USC 1401 (g).

@Justaguy yes. Thanks for bringing it to my attention. I actually noticed after I copied the link that I'd picked the wrong section, but forgot to correct the error before I posted the answer. It's fixed now.