Richard F. Weingroff at the US Department of Transportation, in his article
On The Right Side of the Road, investigated this in the context of why British colonists in what became the USA chose the opposite driving side to their homeland.
He determines that the US practice of driving on the right derives from two colonial practices - driving large vehicles such as the Conestoga wagon and carrying firearms.
He explains that Conestoga wagons were large, wide vehicles driven by two lines of horses, and right-handed drivers typically rode on the rear left horse in order to maintain control (via a whip held in the right hand). Because of the width of the wagons, it was difficult to pass and so drivers seated on the left would choose to pass on the side that allowed them the most visual clearance, which would be on the right.
This practice of driving on the right was codified into Pennsylvania law in 1792 for the Philadelphia-Lancaster Turnpike (present-day US Highway 30). Traffic was affirmatively required by law to keep to the right rather than choosing the most expedient passing side based on context or situation.
Weingroff also observes that the widespread practice of carrying firearms in Colonial America also contributed to traffic passing on the right. Since the majority of the population was right-handed, typical firearm stances would make it easier to shoot slightly to the left than to the right, so people kept to the right in order to prepare to defend themselves against potentially hostile oncoming traffic.