I always opened the car door for the young ladies I dated. It just seemed like “the right thing to do.” After all, my father always opened the car door for my mother – and still does, even after 50 years of marriage (bli e”h). My father was always a “menchlich” kind of person, and I’m sure that’s why he always opens the car door – to honor my mother. It has nothing to do with imitating Gentiles.
(Let it also be known that my wife, even when we were dating, always leaned over after I let her into the car and unlocked my door from the inside.)
(Let it also be known that the Rav of the shule where I grew up – someone who learned in Bais Yosef/Navordok before he escaped from the Holocaust – always opened the car door for his wife. I remember watching very vividly as they would leave shule functions over the years.)