This is sometimes possible but only if the two countries have an agreement to do it and, as far as I know, that's not the case for the Netherlands and Italy in the US.
Such agreements are getting more common but very often it's between neighbouring countries (e.g. the Benelux, Nordic countries) and in places where the destination country does not have any representation. You cannot generally apply to any consulate from any Schengen country.
In this particular case, the Netherlands does have an embassy in Washington and it seems it's not possible to apply elsewhere. You could still inquire at the embassy to see if some other arrangement is possible but it does not look like it.