By most accounts, no. Anne French became the first woman to receive a drivers license in 1900. Girl Scout gave out "Automobile badge" to eligible girl as early as 1916. I remember reading from the 1920-1929 volume of This Fabulous Century that early car companies marketed car to women , especially after the invention of self start so one no longer need to crank the car to start.

Women were never legally forbidden from driving or owning automobiles in the United States. In practice, I suppose, it's entirely conceivable that an individual husband or father would have prevented his wife or daughter from driving, but I don't think that would have been considered the norm. Certainly, female drivers were a large enough market that manufacturers advertised directly to them. The largest obstacle to female drivers was the physical exertion required to start a gasoline automobile before the widespread adoption of the electric starter; for this reason, electric automobiles like the Detroit Electric, Babock Electric, and the aforementioned Baker Electric marketed their cars as easy-to-use alternatives for women well after they were surpassed by gasoline automobiles in terms of performance and price. When the electric starter became standard equipment for gasoline automobiles in the late nineteen-teens, they were immediately marketed to women, often focusing on styling, comfort, and amenities.