Does recognition of the legitimacy of the State of Israel extend to recognition of the legitimacy of any actions taken by the government of Israel on behalf of the State of Israel? Your explanation leaves untouched the question about the relation between legitimacy of State existence and legitimacy of State actions. Those two can be assumed to be the same thing, but at the price of the democratic integrity of the State. It is, of course, better to assume that those two are not the same thing, which begs the question of when objections and resistance to State actions will be defined as rejecting the legitimacy of the State's existence.
Saudi king to meet with Obama on Friday amid Gulf concerns over Iran deal (Reuters)
from the article: It's enough to recognize Israel's legitimacy