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
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.
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
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.