Abolishing ICE without considering the need for those responsibilities and what org, if any, should take them on is certainly risky. But technically, yes, we could abolish ICE and route things ICE does (immigration management, customs enforcement, non-military border security) to other executive departments (not necessarily back to their originating department, since instituting ICE also abolished some of its predecessor departments).
The Programs division [of INS] was responsible for handling all the functions involved with enforcement and examinations, including the arrest, detaining, and deportation of illegal immigrants as well as controlling illegal and legal entry.
[1] https://en.wikipedia.org/wiki/Immigration_and_Naturalization...
https://en.wikipedia.org/wiki/Rebranding#Elimination_of_a_ne...
https://en.wikipedia.org/wiki/U.S._Customs_and_Border_Protec...
>INS, the agency ceased to exist under that name on March 1, 2003, when most of its functions were transferred to three new entities – U.S. Citizenship and Immigration Services (USCIS), U.S. Immigration and Customs Enforcement (ICE), and U.S. Customs and Border Protection (CBP) – within the newly created Department of Homeland Security, as part of a major government reorganization following the September 11 attacks of 2001.
>Prior to 1933, there were separate offices administering immigration and naturalization matters, known as the Bureau of Immigration and the Bureau of Naturalization, respectively. The INS was established on June 10, 1933, merging these previously separate areas of administration.
https://en.wikipedia.org/wiki/Immigration_and_Naturalization...