Assuming that I told you 2 days ago, "I promise I will call you tomorrow". But I didn't. So now I have to apologize you. Then I say "I promised I would call you the next day." is it correct?
Yes, it's correct. Would is the preterite form of the modal auxiliary will. Referring to a past event, you're putting both promise and will in the preterite. And the implicit reference for tomorrow is now: it means the next day in relation to now. Since you're putting the whole thing into the past, it makes absolute sense to do what you've done and to transform tomorrow into something which doesn't have that implicit reference.