Welcome!
Can't I just come back and visit with my passport, as long as I don't exceed six months?
No.
Getting a visitor visa in advance isn't an assurance you can come in and stay for any length of time, and neither is presenting your passport at the border (which you now cannot do because you've been refused a visa). It's all up to the immigration officer who examines your case when you present yourself at the border, whether you have a visa in advance or are requesting one at the border. They can allow, deny, or allow you in for a shorter period of time.
A six month stay as a visitor is exceptionally hard to get, because UK immigration is aware there are few people who can simply take six months off of their life abroad and come to the UK for a visit. You have to have clear ties to your country of residence and most people who have those kinds of ties don't have the time for six months away.
There is no rule to how long you have to leave, again, it's up to the immigration officer on the day. However, a good rule of thumb is to be out of the country as much as you're in it, so you can prove you're not using a visitor visa to reside.
You'll need strong, clear ties to the US and to reapply for a visitor visa. Good luck!
