Before we get to whether "Israel has a right to exist" and why that phrase actually means nothing, let's illustrate with a couple of other common examples. One of the most prevalent and annoying is "family values." Precisely what the hell does it mean to be "against family values"? That's right, it means nothing. But every year another politician or two dutifully trots out the line that s/he is in favor of family values, and no one ever asks them what, exactly, is a family value? What are you supporting? Does anyone recall that this phrase hit the big-time when Dan Quayle -- Dan Quayle of all people -- hurled it at a fictional TV character whose life choices he didn't like (even though they weren't, actually, you know, real choices in a real life that existed or anything)? The fact that we're still being subjected to this nonsense phrase more than two decades later says something -- I don't know what, but something pretty awful -- about the U.S. political scene.
From "the" other "side" of things, we get another bit of political rhetoric you're unlikely to avoid if you ever like to be on the internet, and that is the false dichotomy of gay marriage versus "hate." Gay marriage versus discrimination, yes. Gay marriage versus unequal treatment, certainly. Gay marriages versus people clinging to some misguided ideas, I'll grant you. But all the signs and memes and posters and retweets with phrases like "Do you support gay marriage or do you support HATE?!" are just absurd. What do you hope to accomplish with that? "Yeah, put me down in the hate column." Who says that? (I might add that the idea of being "pro-choice" is also problematic in this way, but then again, that term is fighting against the even more insidious and misleading "pro-life," so the abortion rhetoric battle is long past any hope of real words that mean anything.)
Right, so, Israel. Here's Netanyahu, who doesn't want to
There are three basic problems with Netanyahu's demand.
1. Israel is not the same thing as Jewish people. But this is the emotional reaction they're playing on. They say "Israel's right to exist" and we are supposed to immediately line up on the "right" side, that is to say, the side that equates "not wanting to kill millions of people" with "support for Israel" (to the extent that those three words actually mean anything either). If someone is born, they now exist (note: born) and you don't have a right to kill them. So, that's the end of the story. There is no coda in which their lives, one life or a hundred lives or six million lives or approaching seven billion lives mean any particular nation should or shouldn't exist. It means those one or a hundred or six million or nearly seven billion lives get to be lived. That's what it means.
2. Maybe, just maybe, carving a modern Jewish state out of Palestine was not in fact the best solution. Maybe, just maybe, it caused more problems than it solved. Maybe, just maybe, some people should be willing to discuss this instead of shutting off all debate about it. And yet, instead we sit here demanding recognition of Israel's "right to exist." Does any state have a "right" to exist? Why? Why do any states exist in the first place? For a lot of terrible reasons, for many of them. History, inertia, war, reproduction, heritage, negotiations, genocide, whatever led people to be where they are and establish nation states -- these aren't really the same thing as having rights.
3. It goes back to the oft-repeated but never correctly translated statement that Mahmoud Ahmadinejad allegedly made (except guess what, he didn't) that Israel should be wiped off the map. That loaded phrase in English is an idiom that for some reason everyone takes figuratively and literally when they hear this (alleged but not really made) statement about Israel. But he didn't say it. He was explaining that maybe said modern Jewish state shouldn't have been carved out of Palestine with these particular lines in the sand (and now, on maps) where they are. Kind of like what I'm trying to explain to you all (see #2 above). And that it's OK to have a conversation about this. But we don't have a conversation. We run around ignoring everything except inflammatory rhetoric (real or imagined).
What does it actually mean to recognize the right to exist? What is the point of it? Look at the things around me. My pen. My desk. My cat sleeping on the bed. My comfortable sweatshirt. Do these things have a right to exist? They just exist! You don't have the right to destroy them, but what actual right do they have? What does it mean? It's more like a right to not not-exist by someone else's action. But god forbid we talk nuances or philosophy or, you know, subtlety of thought. We'd rather scream and yell about Israel as if it's a good idea and you're a terrible hater if you don't value it, where "value" is defined to mean "repeat a bunch of cliches and tired old saws that will get a lot of warmongering blood pumping."