In short, no.

To practically avoid enlistment there was no necessity to renounce citizenship. Men abroad, whether legitimately (i.e having emigrated with a passport released by Italian authorities, which presupponed that their status related to military conscription would have been known) or unbeknownst to Italian authorities, were called to arms. For obvious reasons, those who had emigrated in a clandestine manner were expected to show up; the others usually weren't, even if the Italian Army offered them the chance to come back and serve in the war.

The living conditions were pretty miserable and - to be fair - there were close to no extra benefits. During the first months, there wasn't even extra accomodation provided during license, which means that one (de facto volunteer) who had come back from the Americas but had no immediate relations in Italy, would find himself forced to spend the two weeks license living in the streets.

Some of these would later enlist with the US Army, which provided much larger benefits, including the especially hefty insurance prize, roughly equivalent to 50,000 Lire paid in one installment (which was enought to purchase a small parcel of land) - a thing many volunteers would proudly declare to their families as their main reason to go to war.

As for the others, those would have made up a large chunk of the men charged for desertion by the Italian Military Courts (470,000 for not showing up - of which 370,000 living abroad) but these courts were busy with other 400,000 men charged for other violations (including almost 200,000 for desertion). And the military courts and authorities considered more urgent to pursue those men who were actually within their reach, since those were the most directly susceptible to discipline measures, the most suitable as examples, and those who could be sent back to the frone.

In fact the military courts had such a hard time dealing with the many reported violations that proceedings against Italians abroad were substantially shelved. It is also to be noted that, while the cost of a transatlantic voyage was high before the war, by 1915 it could have been prohibitive for many Italian emigrants, who could claim impossibilty to come back as an exhonerating element.

Anyways, there were still roughly 50,000 cases open when in Semptember 1919 the Government of Francesco Saverio Nitti declared a general amnesty (with the exception of some major violations).

This would have cleared anyone living in the US as well - even if frankly it would have been unlikely for their case to be open at all, or considered relevant by the authorities.

As for what your great grandfather might have done, and believed at the time, I really can't say. Italian propaganda originating from the Italian government was close to non existent within the States; with sporadic initiatives left to affluent Italian-American individuals. I am not aware of specific targeting local communities with threats of any kind that would have likely been counterproductive.

Sources - in Italian only -

Monticone, A. - Plotone di esecuzione

Rochat, G. - La grande guerra

Melograni, P. - Storia politica della grande guerra