None of this appears to directly address my point: states can call themselves whatever they want to call themselves. How they hash out their particular identity is not the business of another state or, in the case of the Palestinian Authority, an internationally recognized non-state entity.

I would remind you that there has always been, and still is, disagreement among Israelis, including Zionist ones, as to whether Israel should be a "Jewish state" or "a state of the Jews." The former means that the state itself is concerned with policing a certain specific identity and ethnicity, whereas the latter would mean that the majority of people are Jewish. At present, the Israelis have generally decided on the former. And so, Germany is a state of the Germans, and the US recognizes the Federal republic of Germany. Mexico is a state of the Mexicans and the US recognizes the Republic of Mexico. The State of Israel calls itself the State of Israel and the dominant self-depiction is that it is a Jewish state. But the US just recognizes the State of Israel- and very, very wisely, stays out of the debate as to how those people in that particular state choose to define themselves.

As it happens, I am in favor of the survival of the State of Israel. The US State Dept. is too, and wisely stays out of that debate on their internal self-description. Most other states do, too. Recognizing the state of Israel does not entail endorsing a specific ethnicity. Mahmud Abbas is just following the standard practice of other states and internationally recognized non-state actors in also recognizing that there is a State of Israel, and not endorsing that state's self-depiction.

Actually, all Abbas is saying is that it's not his (or anyone else's) business how the state of Israel chooses to see itself. States recognize states, and they are not typically in the business of telling states what their ethnicity is or ought to be.
The United States has never recognized Israel as a Jewish state, just as it has never recognized Germany as a German state, or India as an Indian state. We are not in the business of ratifying the sort of ethnicity that other states have decided or created for themselves. We just recognize states. That is all. Likewise, no state recognizes the US as the "American state." They are not in the business of agreeing with us on the collective self image we have decided for ourselves.