These days there is a range of drop bar 'road' bikes with a spectrum of capabilities. Using some of Specialized's current range to illustrate:
You can ride a 'race' bike such as the Tarmac with narrow, slick 23mm tires on a gravel or poorly paved surface, but it's not much fun, and you risk puncturing of damaging the wheels.
The Diverge with it's 30mm tire does better, but don't expect to crash it over large potholes without consequence.
The Sequoia with 27.5" wheels and 42mm tires is in mountain bike wheel territory and obviously deals with rough surfaces much better.
Basically bigger tires means better rough surface capability, but at the expense of heavier, harder to accelerate wheels.
For the C&O Canal towpath - a trail I'm very familiar with (fairly smooth crushed gravel, a few ruts and potholes) - a tire 28-35mm with some tread will probably work.
I ran my new bike with 28mm slick tires on the Fairfax County Connector Trail which is far rougher than the C&O Canal towpath trail this weekend with no issues. 28mm on the C&O will be fine, although I still recommend a tire with some tread.