For possession, you say "I have a new car" or "I've got a new car". You do not say "I-have-got"; we use "got" because of the contraction "I've". (Also, you don't say "I've a new car" either.) However, if you add another word, eg. "I've just got a new car", then you are following present perfect grammar rules.

I have a new car - General statement when you say that you have a new car I have got a new car - The moment when you got yourself a new car

I have gotten a new car.