If Israel gets into an actual shooting war with Iran would we get dragged in, as in boots on the ground?
I don't think so.
It would look like every other time Israel and (insert middle east nation/nations here) have gone to war.
So far there have been 6 of them - 1948, 1956, 1967, 1973, 1982, and 2006.
Not to mention virtually continuous asymmetrical activities.
In none of those cases did Americans roll out to boots on the ground to fight on the side of Israel. We've supplied a ton of stuff to back Israel, especially in 1973 which, unlike most of the other conflicts, was actually a very touch and go thing for awhile and Israel got mauled pretty badly.
More importantly, though, the Arab nations around Israel have finally learned that *expletive deleted*ing with Israel is not a sure thing at all. They're well trained, well, equipped, well led, well supported, and they're most often fighting for very concrete reasons -- protecting their nation, families, and homes.
The Arab state armies, on the other hand, generally not as well trained, as well equipped, as well led, as well supported, and when you get down to it, trying to base your military goals in terms of "ALLAH DEMANDS THAT WE EXTERMINATE THE JEW PIG INFIDELS!" just doesn't have the same panache.
Sure, there's the spectre of nukes. Iran doesn't have nukes. Israel likely does. Iran knows that. Iran also knows that, unlike 1948 and 1967, most of the other Arab nations in the region don't exactly view them with Allah's love and compassion. Most of the other Arab nations in the region would, I think, be perfectly happy if Iran were to FAFO and undergo a change of government from the bomb throwing Jihadi fundamentalist group to a more secular government that isn't quite as much of a rabid dog at risk to bite both friend and foe.
Sure, Putin is blustering on about the might of Russia. Ignore him. The Israelis certainly don't put any credence in the ravings of someone whose most successful military enterprise has been to show the rest of the world what the vaunted might of the Russian military is an absolute *expletive deleted*ing joke.