No, there's nothing grammatically wrong with call him the morning of the next day. Arguably, it's semantically ambiguous (as in the old joke 'Call me a taxi.' All right, you're a taxi.') but in reality people accept this to avoid cumbersome phrasing; as Charles said, a native speaker would say call him the next morning. The further away the date is, the more likely it is that on will be used: call me tomorrow (not *on tomorrow), call me the next day or on the next day, but call me on Thursday (?Call me Thursday is used, but is informal).
Your teacher is most likely incorrect.
Old Counterexamples
Some high-profile examples, though dated and not a model for modern grammar, counter the idea that be necessary to may only refer to people.
The Second Amendment of the U.S. Constitution (source):
A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.
Thomas Paine in Age of Reason (source):
But it is necessary to the happiness of man, that he be mentally faithful to himself.
John Locke in A Letter concerning Toleration (source):
Those moral rules which are necessary to the preservation of civil society [. . .].
New Counterexamples
I did a search of COCA for necessary to the [N*] to obtain the following examples. This is not an exhaustive list.
Recent examples include Bloomsbury, Freud, and the Vulgar Passions (1990):
The monetary reforms necessary to the preservation of reasonable price stability are also necessary to the preservation of civilization.
Intermental Thought in the Novel: The Middlemarch Mind (2005):
The physical brain is necessary to the production of a mind [. . .].
"Radical Islam in America" USA Today (2005):
To the contrary, preventing this is necessary to the defense of religious freedom.
American Heritage (2004):
The abdominal operation is necessary to the health of the patient [. . .].
Meet The Press (Spoken) (2011):
Debate that is absolutely necessary to the future of this country in terms of entitlements [. . .].
Microsoft–hardly a paragon of the English language–has an error in Windows 7:
The Selected Disk Is Necessary to the Operation of Your Computer, and May Not Be Cleaned
Examples of the teacher's claim
A COCA search provides a few hits between 1990 and 2010 (source):
He was thinking, from his care in formulating his words, which was more necessary to him with her than it had been with anyone.
What would be more meaningful than being necessary to him in a practical sense [. . .].
She wanted to become so necessary to him that he wouldn't leave her.
She was both intelligent and necessary to him.
I am not so necessary to him as he is to me.
Even if Gorbachev's history was, and still is, necessary to him, it is clearly out of style with many of his successors [. . .].
Note: this only looks at "necessary to him"; further searches with her, them, person names, etc. may yield more examples.
Conclusion
In the OP's case it does sound better to replace to with for; however, the sentences above sound fine to me. My intuitive guess is that the phrase be necessary to (excluding to-infinitives, e.g. "it is necessary to kill") is less common than be necessary for, but I lack evidence to back up that claim. Nevertheless, the teacher's claim doesn't seem founded on real-world usage.
Best Answer
1) is idiomatic English.
2) is possible, but I would understand it with a different meaning, something like "Whenever she answers the phone, she acts responsibly". It is ambiguous (as is my paraphrase) as to whether it is the act of answering the phone which the speakers sees as responsible, or the way she behaves when she has answered the phone.