As for how far will the US go, well in 2002, Congress passed (and Bush signed) the American Service Members Protection Act, more colloquially known as the Hage Invasion Act. It authorizes the president to use all necessary force including invading the Netherlands if an American servicemember or appointed official is ever taken into ICC custody. This includes officials and servicemen of key allies, including Israel.
So will Betanyahu or Gallant actually be arrested? Almost certainly not. The practical effect of this is political not legal.
The goal of protests, boycotting, ICJ applications, ICC warrants, UN (GA and SC) motions, "Undecided" voting in Democratic primaries and so on are to incrementally pressure the two key players here: Israel and, more importantly, the US. Why? Because the US could end the conflict with a phone call. They could end it with a press release.
BDS (Boycott, divest, sanction) movements were considered successful in isolating and ultimately toppling the Apartheid South African regime in the 1970s and 1980s. Given this success, an awful lot of lobbying has been directed at US politicans to pass so-called "anti-BDS" laws that are laws in ~37 states. For example, to be a teacher in Texas, you need to sign a contract agreeing to never participate in a BDS movement against Israel.
So the practical effect of ICC warrants is just to incrementally isolate and pressure Israel.