1: The day Rose left me...
2: The day that Rose left me...
3: The day when Rose left me...
These phrases are noun phrases modified by relative clauses. These are all examples of defining relative clauses (also called integrated or restrictive relative clauses).
In these examples the relative clauses explain which day we are talking about. When we have a defining relative clause like this, we can usually choose either a wh- word or the word that to introduce the relative clause. So in these examples, we can choose either **when* or that.
However, we can also drop the relative word completely, as long as there is a Subject before the verb in the clause. In these examples Rose is the subject of the clause, so we can drop the words when or that.
Compare the following examples:
- That is the man who [Bob punched]
- That is the man who [punched Bob]
In the first sentence if we take away the word who, the relative clause will still have a Subject, Bob. So we can drop the word who here. In the second sentence if we drop the word who there will be no Subject for the verb punched. We cannot drop the word who here.
The word when is almost never the Subject of a relative clause, so we can nearly always drop the word when from defining relative clauses.
Sentences 1-3 are all grammatical. We might have different reasons to prefer one over the others in different specific circumstances. My personal feeling is that in simple sentences it is almost always better to leave out the relative word. In complicated sentences and in technical writing, it is generally better to leave the relative word in. But then this is a question of style, not grammar.
We do not seem to be able to drop the word where in relative clauses, unless the antecedent is the word place.