I've had a baby in the US.
When I discovered I was pg, I registered with an OB/GYN practice - basically made an appointment to see the midwife that they had on staff there. I had loosely registered with a Family Practice when we first arrived (for the kids) but hadn't had any reason to go for myself. Because their maternity care was all OB focussed, I decided to give them a miss.
So, it's perfectly acceptable to make an appointment directly with an OB/GYN practice. The one thing to be aware of is the hospital that they have priviledges at - if you want to deliver at a particular hospital, you are best to choose a primary care practice that has priviledges there.
I found that in Ohio, it was very much like it was in the UK when I had my first in the UK, 10 years earlier. The appointment schedule was once every four weeks until 28 weeks, fortnightly until 36 weeks and weekly thereafter. The offered a scan at around 20 weeks and AFP (!!!) testing at 16 weeks. They also offered GD testing and GBS testing towards the end. I had no problems declining these test, though I am sure they put me down as a weird European.
The big difference was that there was a nurse who did your vitals, and the midwife or OB came into see you after they had these results.
The other thing was that ante-natal care wasn't particularly sociable - at my surgery in the UK, they had AN appts every Tuesday afternoon, and you got to know lots of pregnant mums, plus the parentcraft classes later on. There, you went for your appts at any old time, and were sitting in the waiting room with people going for smears, contraception etc.
The hospital bit for me was good, and I was happy to have my own midwife there who was ready to flout hospital policy. They did treat me as a special case, I think, having come from the UK and having had homebirths there.