A: Can you not tell anyone, please? I’m so sorry about everything. I’m so ashamed of all of it. It’s just… It’s going to be bad and I just need a little bit of time to figure out how I’m going to tell my mom and dad and…
B: Don’t worry. I won’t tell. It’s okay.
I’m just wondering if it is okay to leave out the object (anyone) after the transitive verb “tell” in the sentence “I won’t tell.”
Because I’ve learned some verbs must have an object, for example, a person asks you, “do you like vegetables?”, you can’t just say, “I like”. Instead, you should say, “I like THEM”.
So the verb tell here can be applied to this rule or not?
Typically there would be an object after “tell” if it was a stand alone sentence, but since the object was previously stated it does not need to be repeated. This is more true of spoken English. This does not apply to most verbs. In the example you gave using “like” you would still need to add “them”. As a none native speaker you can never go wrong if you add the object after the verb. The instances where you do not need to add the object are rare and even in those instances adding the object is just as natural. As long as you understand this structure when you hear it I wouldn’t spend a lot of time thinking about how to use it.
“Please don’t tell” is perfectly acceptable. When used like this without any object “tell” has the specific meaning of revealing a secret or, colloquially, “spilling the beans”. Semantically the sense of the object (a secret, e.g.) is implied in the verb, and I guess that’s why it’s not needed explicitly. A more colloquial phrase is “Please don’t blab”. The meaning is the same, but “blab” never (hardly ever?) takes an object; it’s the act of speaking about things that should be left unsaid.
I believe that in the usage “Please don’t tell anyone” that “anyone” is an indirect object. In these examples – “Don’t tell anyone it” or “Don’t tell it to anyone.” – the direct object is “it” and “anyone” is the indirect object.
I’m not sure if it’s “correct” to say “I won’t tell”, but you would be understood in this context. However “I won’t tell anyone” or “I won’t say anything” or even “I won’t” sound more natural if the prompt is “Can you not tell anyone, please?”.
The meaning is slightly different than “Don’t tell.” “Tell” is used when you mean don’t speak of it now or in the future to someone. If your brother sees you doing something that you shouldn’t do, you say “Don’t tell Mom!” whenever he sees her.
“Say” is used in these sorts of examples when you mean right now. “Say” means speak, utter. If you’re with two other persons and you fear one is about to say something to the other that the other shouldn’t know, you interject “Don’t say it!” It’s also used between two people, e.g. you’re doing something obviously wrong and ruin your work. Before your companion can utter a reproach you say, “Don’t say it”, meaning that he doesn’t have to say what you did wrong, it’s obvious and saying anything about it won’t help.
[edit] Back to the first example, you could also use “When you see Mom, don’t say what I did.” That still has the immediate sense as it applies to that specific time when he sees Mother.