It absolutely does not make me feel uncomfortable.
1. Historically, I would say that the Zionist settlers had no right to move to a new land and terrorise the locals to establish their state. So from a historical context, no the State of Israel does not have a right to exist.
2. Presently, even if the Palestinian issue were solved and the '67 borders for Palestine and Israel were made manifest and Jerusalem made an international city or divided between the two states, I still would have a problem with the identity espoused by the medina.
3. I realise both these two positions are both considered quite extreme. Although in the חברה in which I roll, it's quite normal. (For reference, the חברה to which I refer includes Jews, Christians and Muslims.)
4. [dohtml]
I find this relatable.[/dohtml]
5. All that being said, if the medina somehow changes its present M.O. so significantly that it allows for a Palestinian neighbour on equal terms, that satisfies the majority of Palestinians, I will, not as a Jew (see #2), but as a person, accept its right to exist.
6. This opinion is quite normal in my משפּחה and, בּרוך השם, is increasingly mainstream in American Jewish thought.
7. These are mainly political beefs, and my first thoughts of rage at the medina were triggered by a political/humanitarian impulse. It was the most recent siege of Beirut. When it happened I remember thinking how incredibly hostile that seemed, and how it almost seemed hostile enough to explain anti-Jewish sentiment among Arabs. Indeed, it was then that I realised that the medina's present incarnation could not be a force of peace or good in the region, and was probably doing more harm than good to Jews of the world. (Lately I have begun to consider that it consciously intends to foster Judeophobia among the worlds Muslims in general and Arabs in particular, so as to rally Western guilt to their cause and have more targets to prove the lack of civility that the civilisation[s]/culture[s] Palestinians belong to engender.)
8. Religiously, it does bother me that the state exists in spite of messianic belief, but I am not Orthodox enough for this to be my principal beef. I would support some kind of Jewish state if its identity were different, but I can't endorse a religiously authoritarian state proclaiming to be the representative of world Jewry doing so in direct violation of the strict rabbinical Judaism they claim to represent.
9. I certainly don't trust the American Christian Right any further than I can throw them. I usually avoid trusting anyone's Right. I trust Wright though. Wright was right.
So I hope that answered all your questions, but it may have provoked some new ones.
PS: Sorry for all the Hebrew shit, but this provoked (for obvious reasons) a Jew-y mood. You're lucky I just used words from Hebrew and didn't write full Yiddish sentences.

For reference:
Crew=חברה
Family=משפּחה
Thank G-d=בּרוך השם
And although I wrote it in Roman characters and like to pretend it's an English word, I should also note that "medina" (מדינה) is "state" in Hebrew.