Is it right that "to adults" is obligatory as in "I taught English to adults"?

"I taught English" is a complete, correct sentence. "I taught English to adults" is also a complete, correct sentence.

Native speakers of English are likely to understand "I taught English" as equivalent to "I taught English to school-aged children", because that's the default assumption for anyone teaching anything. If you say either sentence in a country where English is the common tongue, native speakers will assume you mean you taught English literature and/or writing skills, not the language, because that's what the school course called "English" will focus on.

In comments on the other answer, you seem to be insisting on a grammatical distinction between a "complement" and an "adjunct". English is not a language whose constructions fall into well-defined categories with sharp edges. Prepositional phrases like "to adults", "on the table", etc are especially hard to categorize. I think it's easier to understand your examples in terms of the transitivity of their main verbs. English has at least five different transitivities for verbs:

• I put the dish on the table: ditransitive. Subject, indirect object, and direct object are all required. You have to say what you put and where you put it. "On the table" is a prepositional phrase, but any locative construction will do.

• I taught English: transitive. Subject and direct object are required. You have to say what you taught, unless it's clear from context. You can, again, add more information with prepositional phrases.

• I ate [an apple]: ambitransitive. Subject is required, direct object is optional. You can say what you ate, using a direct object, but you don't have to. (Perhaps you only want to communicate that you're not hungry right now.)

• I slept: intransitive. Only a subject is required. You have to say who slept, but no more words are required for a complete sentence. You can add more information with prepositional phrases, e.g. "at home", "on the train", "for only three hours".

• It rained: impersonal. Not only are no object phrases required for this complete sentence, the "it" isn't a real subject. You can't leave it out, but it has no referent and you can't substitute any other subject phrase without sounding weird. (Many of the world's languages treat verbs describing weather this way.)

(It is not clear to me that any language's constructions fall into well-defined categories with sharp edges, but that's a whole 'nother can of worms.)

If a verb requires a subject or a direction then you must include it otherwise the sentence is not complete, for example:

"She put the dish..."

This is not correct because there is no direction specified for the verb "put". You could say "put the dish down" or "put the dish on the table" which would indicate either a place or a direction.

In the example in your question, "taught" is your verb, but you can teach a subject or you can teach a person, so the sentence is complete with either the name of the subject (English) or the person(s) you are teaching (the adults):

I teach adults

OR

I teach English.

Including both is optional and depends on how much detail you want to convey.

If you need to indicate that you were teaching adults as opposed to children then you do need to say something like:

I taught English to adults.

OR

I taught adults English.

The preposition is not required here because the relationship between the nouns is clear. You could not say...

I taught English adults

... because that sounds you are using "English" as an adjective to describe the adults' nationality.

Because "English" is the name of the subject being taught there is no need for a verb, but if you introduced a verb into the sentence you would need the preposition again:

I taught adults to speak English.