Some Suggestions
So first off, I am obviously not someone who knows a lot about you or your friend, nor do I know the specifics of your situation, so please, take everything I say with a grain of salt, as my advice will be very general.
Now, without further ado, my two cents on the matter: Honestly ask yourself, what is the argument over? Does this disagreement go against your moral beliefs? If so, be serious with your friend and explain why (note: moral beliefs are something that you feel very strongly about, and it can be very hard to have friends who share different or opposing moral values). If this is not a moral difference issue, is the argument something you could change your own views on? The reason I ask, no matter how much pondering you do, you can’t change someone who doesn’t want to change, so always start with yourself.
Also, it is good to keep in mind when disagreeing with a friend, listening and talking equally is extremely important. One person cannot expect the other to listen if they aren’t doing it themselves. Check yourself, are you listening to why your friend thinks you are being immature? Do you comprehend how they want you to ‘grow up’? Next, check your friend, (and be real with yourself) are they listening to you? Are they comprehending your side of the argument as much as you? Both participants need to listen to and comprehend the others words in order for both to come to a standing agreement. In the case of unwillingness to listen, reevaluate how important the friendship is, and decide how much time/energy you are willing to spend trying to (KINDLY) explain your point of view to your friend.
To sum all up, decide how important the friendship is, decide what you are willing to change and ask the same of your friend. Try to reach a compromise that solves the issue while preventing future ones by listening to one another. And do please remember, this is just my opinion.