I think Israel is a Western Democraty for these reasons:
"Western", due to its cultural ties and since it shares western values.
"Democracy" because of many institutions, including the rule of all, the free elections and government form, minority rights protection. It is true that it's not perfect, but overall the big picture is pretty clear, as far as I can see.
To elaborate: in my opinion, Israel is a new democracy, it definitely has some deviations from the old western democracies, but these are relatively small to change the overall spirit of the country and the form of government (It is a bit like saying that US is not a western democracy because it has an electoral college.) What more there are part of the ongoing political debates and changing norms.