I'm a native born American currently living in Israel as a citizen (dual citizenship).
Can I leave Israel to travel to Europe with only my Israeli passport, or is my US one a 'must have'?
To exit Israel, all you need is your Israeli passport. Israelis are uniquely granted visa-free access to all of Europe.
So if Europe's where you're going, your US passport will not be needed
There's no requirement for you to take your US passport with you on a trip to Europe. However, it is a good idea to do so. For example, you might have a sudden need to travel unexpectedly to the US, such as a friend or family member falling seriously ill, and doing that will be far more complicated if you don't have your US passport with you.