Can a parent use dual citizenship to transport a child out of the USA without a US passport?

The US State Department recommends informing the embassy of the other country (in this case Italy) about the situation. Your best bet, I would think, would be to make sure a note is in the airlines’ database for the Italian passport, with the hope that they would not allow her to board with it.

I’m sure there are all sorts of rules about children crossing borders that don’t apply to adults, but in practice, most countries, the U.S. included, aren’t very picky about the documentation of adults LEAVING unless they are on the lookout for a fugitive. And the U.S. does NOT require anyone, regardless of age, to have a passport in order to leave. They might stop the kid, but the passport wouldn’t have anything to do with it. Needing permission to ENTER could reasonably be considered one of the defining characteristics of a what is called a country, but needing permission to LEAVE is clearly characteristic of a police state.

Yes, if your spouse succeeds in acquiring a non-US passport for your child, it is very unlikely they will be prevented from exiting the US without approval from the other parent.

If the child has a non-US passport, including an Italian one, there is nothing to prevent the child exiting the US and flying to Italy, either directly or via another country. Italy imposes restrictions on children with an Italian passport travelling without a parent, e.g. on Alitalia’s site:

Minor accompanied by parent/legal guardian – the name of the parent or
legal guardian traveling with the minor must be on the travel document
and a birth certificate or certificate of family status must also be
carried so that the authorities can ascertain the identity of the
child’s parents

This will not be of any help if the child is travelling with a parent.

This US government site gives advice regarding the question:

Your child, although they were born in the United States or abroad to
a U.S. citizen parent, might also be a citizen of another country. A
child may acquire another nationality without the consent of the U.S.
citizen parent.

A child may acquire citizenship by the child’s birth abroad, by a
parent born outside the United States, or a parent who acquired a
second nationality by naturalization.

Enrolling in the Children’s Passport Issuance Alert Program does not
stop a dual national child from getting (or traveling) on a foreign
passport. If your child has, or might have, another nationality,
contact the country’s embassy or consulate directly to ask about
denial of that country’s passport.

Following a comment from @user102008, it’s not legal to apply for an Italian passport for a child without the consent of both parents (absent special circumstances):

a passport application using the downloadable form below signed by: a)
both parents, or b) by one of the parents with the signed consent of
the other parent and a photocopy of the latter’s identification
document;

My child has travelled to and from the US with only one parent and has entered the US on a US passport and exited the US on a non-US passport and has not required any documentation from the non-travelling parent.

If the child has a foreign passport, there’s not much chance that anyone would stop the child from traveling overseas without a US passport. The US government does not enforce the law requiring US citizens to bear a valid US passport when leaving the country.

However, many countries have measures in place to prevent international child abduction, and they will not allow a child to enter with just one parent unless the parent has proof of sole custody or a notarized letter of permission from the other parent. Airlines know about these rules and enforce them. These rules often operate independent of the child’s nationality. Even those that do take nationality into account, for example Italy’s, may also protect the rights of the foreign (i.e., US) parent. See for example Lufthansa’s page on the topic:

Unaccompanied minors and children under 14 years of age with Italian citizenship, who are travelling alone or with someone other than a parent or legal guardian, must carry the following completed declaration form: ‘Dichiarazione di affido’ (Law no. 1185/67). The form can be obtained from the local police station (‘questura’), but not from the airport police. The form must be fully completed and must be carried together with the child’s passport. If the documentation is incomplete, the child may be refused boarding.

You can find further information on the website of the Polizia di Stato.

There’s no mention of the parents’ nationality. More applicable to the case mentioned here:

Unaccompanied children and teenagers under the age of 18 travelling to/from an airport in the USA, and unaccompanied children and teenagers under the age of 19 travelling to/from an airport in Canada, require a notarized consent letter from their parents or legal guardians. If a child or teenager is travelling with only one parent or guardian, the notarized consent letter from the parent/guardian who is not travelling with them is required. If a parent/legal guardian has sole custody, then proof of sole custody must be presented. Many countries demand a consent letter from the parents or legal guardians. Children and teenagers without notarized consent letters may be refused boarding.

Please contact the nearest consulate of the country to which you are travelling for information regarding additional documents that may be required. Further information is also available at:

• For the USA www.travel.state.gov
• For Canada www.travel.gc.ca