Initially, they are both correct. However, they have different meanings, and one of them requires more of a sentence.
“I’m done.” Is “I am done.” The grammar is” Subject + state of being verb + adjective. Yes, “done” can be an adjective. It is a past participle of the verb “do,” and past participles can be used as adjectives. It means “I am finished.” “Finished,” in this sentence, is also an adjective, a past participle style adjective. In this sentence, the adjective describes the subject, and the state of being verb links the two. Another example a sentence with this kind of grammar is:” I am tired.” “Tired” is the past participle used as an adjective to describe the subject. If the adjective is not the past participle, here is an example: “I am hungry.” In this sentence, it is still “subject + state of being verb + adjective. “Hungry” is the adjective that tells about “I,” the subject. Another example would be “She is pretty.” Again, this is “subject + state of being verb + adjective.” “Is” is the state of being verb that links the words, “she” is the subject, and “pretty” is the adjective that describes “she.”
As for the other expression: “I’ve done,” it is “I have done,” and it actually is incomplete because “do” is a transitive verb. There are two kinds of verbs, transitive and intransitive. Transitive verbs need a direct object, and intransitive verbs don’t need a direct object. A direct object is a noun or pronoun that receives the direct action of the verb. You find out the direct object by asking “what?” Here is an example of a transitive verb: “I read the book.” Ask “what do you read?” The answer is, “I read the book,” so “book” is the direct object. If you have a verb like “go,” “go” can’t take a direct object, so it is intransitive. The only think you can do with go and tell who goes, where the go, and when they go. There is no question like “what?” there. “Do” just happens to need a direct object, so it is transitive. Every language has verbs like this, not just English. If you see “I’ve done,” it is incomplete. It needs a direct object. If you don’t know what the direct object is, you can just put “it” as the direct object.
“I’ve done it.” or “I have done it.” “I” is the subject. “Have done” is the verb. “It” is the direct object. “Have done” is a present perfect verb. that means it begins in the past and continues until the present. It is perfect or finished in the present, but it is considered a past tense verb. To make a present perfect verb in English, you need to have either “has” or “”have” and the past participle of the verb. Does that sound familiar? There is more than one use for the past participle of the verb. In the first expression, it was used like an adjective, but in this expression, it is part of the present perfect verb.
You ask, “Which one is correct?” If you leave the “it” or direct object out, only “I’m done” is correct. If you add the direct object to the second expression, then it is also correct.