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'?
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.