Your sentence is grammatically incorrect in the first place. So let's try to fix it before we move on. You sentence currently has two clauses, separated by a comma:
- The relative direction you were born within the map of your home country
- that is where you will find your inner peace in the same direction of the world map.
Your first clause doesn't have a verb, so that is the first problem (otherwise you sound like Yoda from Star Wars). My suggestion, therefore, would be to rearrange the two. Here's what I have come up with for fixing the sentence.
The same direction on the world map as that of the relative direction of your country from which you were born is where you will find your inner peace.
This solves the problem of both the grammar and the usage of the word "that" as a pronoun. "That" can be used as a pronoun, however. As a pronoun, it indicates a specific thing that is either assumed from context or from previous experience. If you said "That was fun!" to a friend, then it is usually assumed that whatever you two just did together was the thing referred to by "that." You could also say "Remember when we went on the roller-coaster? That was fun.", where the "that" refers to the activity of "when we went on the roller-coaster."
In summary, "that" is a pronoun, and you were more correct than your friend. Be aware however, that your original sentence is grammatically incorrect, and that it may need to be reworded or replaced with a more coherent/effective saying.