They don't deny that placing "Israel" on a passport might cause some adverse reaction from Arab states. Arab countries are very sensitive to the word "Israel" on passports; Americans who travel widely in the Middle East are routinely issued two passports, because anyone presenting a passport stamped by the Israelis will be denied entry to a number of Arab countries. Nor is it just Arab countries who scrutinize passport entries: the Chinese government once threatened to refuse entry visas to Americans whose passports showed them as born in "Taiwan," because that might imply recognition of "two Chinas." (That dispute was smoothed over, but illustrates the potential.)
These arguments don't seem very strong. Discrimination is the essence of foreign policy, and asking the Supreme Court to hold that a specific act of policy will have what Zivotofsky calls "negligible or trivial impact" on American foreign policy seems like an invitation to go where even self-confident Justices fear to tread.
In its amicus brief, a group of members of Congress ask the Court to ignore the "political question" issue and redefine the foreign relations power, holding, in particular, that Congress has "plenary authority over passports and documentation of birth abroad."
From a transcript of Monday's argument, it seems to me that the Congressional argument is the only one that has a chance of winning. And that chance seems slim. What the Court would have to hold is that Congress can determine any instance of American foreign policy by statute. As Justice Scalia asked Nathan Lewin, Zivotofsky's lawyer, "It seems to me you are not arguing for a co-equal congressional power, you are arguing for a superior congressional power." Lewin was reluctant to confront this aspect of the case. He argued that what Zivotofsky was asking was "trivial ... all that happens with this statute is the 50,000 American citizens have the same passport as 100,000 other American citizens who were born in Tel Aviv or Haifa." They need that because "they are being denied a certain sense of self-respect that they feel they should be able to have in terms of their own identification."
Chief Justice Roberts asked, "So we are supposed to decide whether or not the executive is correct in saying that it's a significant problem. And he says that, but we know foreign policy better; we don't think it's a big deal." Could a decision for Zivotofsky really be cabined to "trivial" matters or issues of self-respect? What, Justice Scalia asked, if Congress voted to allow passport applicants to list their birthplace as "Israel, the only democracy in the Middle East"? Lewin said Congress would have the power to do that.
Solicitor General Donald Verilli, arguing for the government, told the Court that specifying place of birth was "an exercise of the Executive's exclusive recognition power." After some back and forth about what the text actually says (spoiler: the word "exclusive" isn't in there, nor is "recognition"), Verilli qualified: "If the reception clause were not in the Constitution -- but we had the same history that we have now and the same functional considerations about the need for it being in the control of the executive," the government would still argue that the power was exclusive to the president.