Flying with my daughter next week to Israel.
She was born in the US, has currently only Israeli passport. Her passport does have a US visa of course. We're nonpermanent residents.
So I have to go and urgently get her a US passport?
There is a legal requirement for US citizens to have a US passport when entering and leaving the US, though there are some exceptions. (The exceptions mostly concern other kinds of documents that are acceptable when traveling by land or sea to Canada, Mexico, or the Caribbean.) The law is 8 USC 1185(b):
Except as otherwise provided by the President and subject to such limitations and exceptions as the President may authorize and prescribe, it shall be unlawful for any citizen of the United States to depart from or enter, or attempt to depart from or enter, the United States unless he bears a valid United States passport.
There is no penalty for violating this law. And, of course, US citizens have an inherent right to enter the US. In practice, therefore, if a border officer recognizes that you are a US citizen without the required documentation, they are supposed to inform you of the requirements and then waive the requirement of section 1185(b) so you can enter.
For more discussion, see What is the penalty for US citizens entering/leaving the US on a foreign passport? on travel.stackexchange.com.
There was formerly a regulation allowing US citizen children of foreign diplomats to travel to the US on their foreign passports. This was removed in connection with the Western Hemisphere Travel Initiative. It was at 22 CFR 53.2, which read as follows in 2006:
(e) When he is under 21 years of age and is a member of the household of an official or employee of a foreign government or of the United Nations and is in possession of or included in a foreign passport;
However, in 2007, paragraph (e) concerned the NEXUS program, and the paragraph concerning children in the household of foreign officials and employees was absent. The change was published in the Federal Register on 24 November 2006.
The Department of State's [Foreign Affairs Manual] indicates that they continue to follow this policy despite the change in regulations; at 9 FAM 202.1-2 VISA-RELATED ISSUES WTIH U.S. CITIZENS, item (c) appears to have been last modified in 2011:
c. Applications for Visas for Certain Dual National Children:
(1) You should advise parents who apply for visas for dual national children that regulations prohibit the issuance of a visa or other documentation to a U.S. citizen or national for entry into the United States as an alien. The children of foreign government officials, however, may use their foreign passport for entry into the United States.
(2) After the U.S. citizenship of a child has been determined by a citizenship officer, the consular officer may, to avoid delay or difficulty, give a written statement to the parents for presentation to carriers or immigration officials. The statement should make clear that the bearer of the foreign passport is a dual national child of a foreign government official or employee who is traveling to the United States on official business and as such may enter the United States on the foreign passport as an exception to the provisions of INA 215(b) regarding valid passport requirement.
(3) A child under 12 years of age who is included in the passport of an alien parent in an official capacity may be admitted if evidence of U.S. citizenship is presented at the time of entry. A determination of the childs citizenship should be made by citizenship officer prior to departure from a foreign country and the parent should be instructed to have evidence of such citizenship available for inspection by the admitting Department of Homeland Security Officer.
If this is indeed how your daughter got her US visa then the State Department's willingness to issue the visa presumably implies that CBP should allow her to enter with her Israeli passport and that visa.
The Dept. of State distinguishes primary evidence of citizenship, which includes a valid (could be expired) US passport, or an original or certified copy of a birth certificate if it has all of the required information. This is used, however, for applying for a passport. One can use secondary evidence of citizenship, according to the Dept. of State. In either case, State does not actually determine who is admissible: Homeland Security makes that determination. There is an internet meme saying that "All U.S. citizens including children must present a passport or other approved travel document when entering the United States", but none of that comes from official websites. Customs and Border Protection says nothing informative about the topic. Although technically a US citizen cannot be denied admission to the US once they have arrives, nothing addresses the matter of whether a person arriving with a certified birth certificate could use that to gain admission, i.e. there is no evidence that that would suffice.
This question is largely rendered moot because airlines require that passengers have acceptable travel documents, thus they probably won't let you on with just a birth certificate.
According to US law, someone who is an American citizen can only travel in and out of the US with their US passport.
Because your daughter was born in the US, she is automatically an American citizen, and therefore is legally obliged to obtain a US passport before leaving and re-entering the US.
Now, there might be a difference between "legally obliged" and obliged in actual practice (how would they even know without your documentation?), but I don't have any experience there, so I'm not going to comment any further than that.