Is it okay to omit the object in this sentence?

Question

Is the following sentence grammatically correct:

He left me without informing.

Answer 1

Is it okay to omit the object?

What object?

If you use informing in an object verb sense, then an object must be supplied or it is nonstandard grammatically and has no sensical meaning in any ordinary context.

It is almost certainly--or almost always--nonstandard or nonsensical to write or say He left me without informing.

If we deem informing to be a noun, it is difficult to think of a context in which the sentence would make sense. He left me without purple follows a common grammar pattern, but is also generally nonsensical.

Deeming informing an intransitive verb seems to give the construction the best chance at attaining sentencehood. However, it's challenging to contrive a situation in which a sentence makes any sense.

The relevant definition would likely be to provide confidential or incriminating information to an authority, since the no-object definition of providing information is apparently only or almost only used in passive constructions (She was informed of the incident).

We might come up with something like:

What kind of person is he? The kind who doesn't inform on anyone. He had a chance to inform on x during breakfast. He finished breakfast without informing. He might have informed on y at lunch. He finished lunch without informing. He had chances to inform all day long. He never did inform. He was asked to inform twice more before he joined me for dinner, and three more times while he was with me. He left me without informing.

Beyond a similar sort of contrivance, we can't meaningfully use the string of words as a sentence.

In short, again, essentially, no!

Answer 2

You ask whether the gerund^† informing may be employed as a bare noun, without the complements proper to its use as a verb:

He left without informing.

You suggest that this use is licensed by analogy with the unquestionably acceptable

He left without warning.

Gerunds, however, sit right on the border between noun and verb, and each gerund carves out its own territory on either side.

Warning, for instance, has a strong presence on the noun side of the border. It has uses which exhibit the defining characteristics of a noun:

• Morphologically, it may be inflected for plurality and possessive case.
• Syntactically, it may act as head of a noun phrase, modified by determiners and adjectives, and with or without these dependents it may act as a complement (Subject or Object or Predicative Complement) in clause structure or as Object of a preposition.

  The Weather Service has issued three urgent warnings since 9 am.
  The level of the warnings has increased from amber to red.
  The warnings’ effect has been to empty the streets and shops.

But even as a frank noun the gerund warning may take some complements of the underlying verb:

Their warning us that a tornado is imminent has sent us to our basements.

And idiomatically one or more of those complements may be obligatory. For instance, warning in your example "He left without warning" can only signify a warning that he was going to leave. With any other reference, even if that is wholly recoverable from the discourse context, you must supply at least the Indirect Object.

Q: Did he warn you that he was going to press charges?
A: No. He left without warning us*.

Informing has a much weaker presence as a frank noun. By way of illustration, take a look at this Google Ngram, which demonstrates how rarely informing is used with an article:
(Moreover, if you dig into the underlying hits on {a/the} informing you will find that in most, perhaps all, informing acts as an adjective, not a noun.)

And informing, like warning, has idiomatic constraints on its nounier uses. In the case of informing the principal constraint is the contrast between transitive and intransitive senses. Most uses of inform are transitive: we inform somebody of a fact, we inform substance with meaning. But inform also has an intransitive sense: “give information to authorities” on or against someone—information, that is, about the someone’s criminal activity:

Peachum: Inform against him and hand him over to the gallows, and you are at once a rich widow. — The Beggar’s Opera

Consequently, informing can only be used without complements in this intransitive sense; with other meanings you have to provide at least the recipient of the information, just as you do with warning:

He left without informing = He left without betraying his comrades,
BUT
Q: Did he he tell you what he was going to do?
A: No, he left without informing me.

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^† _{gerund: the ‘present participle’ or ‘active participle’ or -ing form of a verb employed as a nominal—that is, in a syntactic role characteristic of a noun.}