On the one hand, Pio Nono is the kind of person to take himself and his office seriously enough to actually do such a thing. Not actually pursue a war, of course (how?), but consider himself at war and call on the faithful to take action. Given the relatively freewheeling attitude towards individual rights, tyranny of the majority and mob justice then prevailing in the USA, that could get very unpleasant in short order. Kulturkampf, but with more lynchings and shootings.
On the other hand, this must be in the running for the top ten dumbest things to do, so I can't see Washington starting it.
Once it takes off though - what repercussions for the US attitude towards France (now not only meddling in Mexico but upholding an avowed enemy), Italy (putatively an ally) and Germany (who, after all, took France out of the equation, allowing a modern, secular Italy to remove a belligerent enemy of the United States)? Will this hold? Most of Europe will see the whole affair as tragicomedy, of course.
Washington will certainly not extend its (apocryphal?) offer of exile to Pius IX after 1870. What of the American bishops? Will their concerted opposition at Vatican I be viewed as a sign of divided loyalties? If the pope tries to remove the lot from office, he could lose the entire American Catholic church. I'm sure WAshington would take as sympathetic a view of these communities as Berlin did of the Altkatholiken. Any changes down the line for immigrant communities pre- and post-1870? Might the Irish be considered patriotioc American Catholics and the Italians dangerous Ultramontanists (Ultramarists?)?
Of course, you could also just detail the US Navy to blockade Ostia and see what the Mediterranean squadron has to say on the matter. One suspects rather firmly "no, bad USA, you mustn't!"