When I walked into the art gallery, the first thing I saw was a notice that said: Don't touch.

Was is a predicate verb in the main sentence and cannot be deleted. It can't be deleted because it replaces the advance notice as the subject in the attributive clause.

The structure of this sentence: firt time is the subject of the main sentence, and what I see is the attributive clause behind it, when leads the adverbial clause of time, was is the predicate, a notice is the predicative, and which means ... is the attributive clause.

When I entered the gallery, the first thing I saw was a sign that said "Don't touch".